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<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html>
<head>
<!-- Generated by HsColour, http://code.haskell.org/~malcolm/hscolour/ -->
<title>Math/Combinatorics/GraphAuts.hs</title>
<link type='text/css' rel='stylesheet' href='hscolour.css' />
</head>
<body>
<pre><a name="line-1"></a><span class='hs-comment'>-- Copyright (c) David Amos, 2009. All rights reserved.</span>
<a name="line-2"></a>
<a name="line-3"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Combinatorics</span><span class='hs-varop'>.</span><span class='hs-conid'>GraphAuts</span> <span class='hs-layout'>(</span><span class='hs-varid'>isVertexTransitive</span><span class='hs-layout'>,</span> <span class='hs-varid'>isEdgeTransitive</span><span class='hs-layout'>,</span>
<a name="line-4"></a>                                     <span class='hs-varid'>isArcTransitive</span><span class='hs-layout'>,</span> <span class='hs-varid'>is2ArcTransitive</span><span class='hs-layout'>,</span> <span class='hs-varid'>is3ArcTransitive</span><span class='hs-layout'>,</span> <span class='hs-varid'>isnArcTransitive</span><span class='hs-layout'>,</span>
<a name="line-5"></a>                                     <span class='hs-varid'>isDistanceTransitive</span><span class='hs-layout'>,</span>
<a name="line-6"></a>                                     <span class='hs-varid'>graphAuts</span><span class='hs-layout'>,</span> <span class='hs-varid'>incidenceAuts</span><span class='hs-layout'>,</span>
<a name="line-7"></a>                                     <span class='hs-varid'>graphIsos</span><span class='hs-layout'>,</span> <span class='hs-varid'>incidenceIsos</span><span class='hs-layout'>,</span>
<a name="line-8"></a>                                     <span class='hs-varid'>isGraphIso</span><span class='hs-layout'>,</span> <span class='hs-varid'>isIncidenceIso</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-9"></a>
<a name="line-10"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Either</span> <span class='hs-layout'>(</span><span class='hs-varid'>lefts</span><span class='hs-layout'>)</span>
<a name="line-11"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>List</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>L</span>
<a name="line-12"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Map</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>M</span>
<a name="line-13"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Set</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>S</span>
<a name="line-14"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Maybe</span>
<a name="line-15"></a>
<a name="line-16"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Common</span><span class='hs-varop'>.</span><span class='hs-conid'>ListSet</span>
<a name="line-17"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Core</span><span class='hs-varop'>.</span><span class='hs-conid'>Utils</span> <span class='hs-layout'>(</span><span class='hs-varid'>combinationsOf</span><span class='hs-layout'>,</span> <span class='hs-varid'>pairs</span><span class='hs-layout'>)</span>
<a name="line-18"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Combinatorics</span><span class='hs-varop'>.</span><span class='hs-conid'>Graph</span>
<a name="line-19"></a><span class='hs-comment'>-- import Math.Combinatorics.StronglyRegularGraph</span>
<a name="line-20"></a><span class='hs-comment'>-- import Math.Combinatorics.Hypergraph -- can't import this, creates circular dependency</span>
<a name="line-21"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Algebra</span><span class='hs-varop'>.</span><span class='hs-conid'>Group</span><span class='hs-varop'>.</span><span class='hs-conid'>PermutationGroup</span>
<a name="line-22"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Algebra</span><span class='hs-varop'>.</span><span class='hs-conid'>Group</span><span class='hs-varop'>.</span><span class='hs-conid'>SchreierSims</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>SS</span>
<a name="line-23"></a>
<a name="line-24"></a>
<a name="line-25"></a><span class='hs-comment'>-- The code for finding automorphisms - "graphAuts" - follows later on in file</span>
<a name="line-26"></a>
<a name="line-27"></a>
<a name="line-28"></a><span class='hs-comment'>-- TRANSITIVITY PROPERTIES OF GRAPHS</span>
<a name="line-29"></a>
<a name="line-30"></a><a name="isVertexTransitive"></a><span class='hs-comment'>-- |A graph is vertex-transitive if its automorphism group acts transitively on the vertices. Thus, given any two distinct vertices, there is an automorphism mapping one to the other.</span>
<a name="line-31"></a><span class='hs-definition'>isVertexTransitive</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-32"></a><span class='hs-definition'>isVertexTransitive</span> <span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span> <span class='hs-comment'>-- null graph is trivially vertex transitive</span>
<a name="line-33"></a><span class='hs-definition'>isVertexTransitive</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>orbitV</span> <span class='hs-varid'>auts</span> <span class='hs-varid'>v</span> <span class='hs-varop'>==</span> <span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span> <span class='hs-keyword'>where</span>
<a name="line-34"></a>    <span class='hs-varid'>auts</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts</span> <span class='hs-varid'>g</span>
<a name="line-35"></a>
<a name="line-36"></a><a name="isEdgeTransitive"></a><span class='hs-comment'>-- |A graph is edge-transitive if its automorphism group acts transitively on the edges. Thus, given any two distinct edges, there is an automorphism mapping one to the other.</span>
<a name="line-37"></a><span class='hs-definition'>isEdgeTransitive</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-38"></a><span class='hs-definition'>isEdgeTransitive</span> <span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span>
<a name="line-39"></a><span class='hs-definition'>isEdgeTransitive</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-layout'>(</span><span class='hs-varid'>e</span><span class='hs-conop'>:</span><span class='hs-varid'>es</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>orbitE</span> <span class='hs-varid'>auts</span> <span class='hs-varid'>e</span> <span class='hs-varop'>==</span> <span class='hs-varid'>e</span><span class='hs-conop'>:</span><span class='hs-varid'>es</span> <span class='hs-keyword'>where</span>
<a name="line-40"></a>    <span class='hs-varid'>auts</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts</span> <span class='hs-varid'>g</span>
<a name="line-41"></a>
<a name="line-42"></a><a name="-%3e%5e"></a><span class='hs-definition'>arc</span> <span class='hs-varop'>-&gt;^</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varop'>.^</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-varid'>arc</span>
<a name="line-43"></a><span class='hs-comment'>-- unlike edges/blocks, arcs are directed, so the action on them does not sort</span>
<a name="line-44"></a>
<a name="line-45"></a><a name="isArcTransitive"></a><span class='hs-comment'>-- Godsil &amp; Royle 59-60</span>
<a name="line-46"></a><span class='hs-comment'>-- |A graph is arc-transitive (or flag-transitive) if its automorphism group acts transitively on arcs. (An arc is an ordered pair of adjacent vertices.)</span>
<a name="line-47"></a><span class='hs-definition'>isArcTransitive</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-48"></a><span class='hs-definition'>isArcTransitive</span> <span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span> <span class='hs-comment'>-- empty graphs are trivially arc transitive</span>
<a name="line-49"></a><span class='hs-definition'>isArcTransitive</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>orbit</span> <span class='hs-layout'>(</span><span class='hs-varop'>-&gt;^</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span> <span class='hs-varid'>auts</span> <span class='hs-varop'>==</span> <span class='hs-varid'>a</span><span class='hs-conop'>:</span><span class='hs-keyword'>as</span> <span class='hs-keyword'>where</span>
<a name="line-50"></a><span class='hs-comment'>-- isArcTransitive g@(G vs es) = closure [a] [ -&gt;^ h | h &lt;- auts] == a:as where</span>
<a name="line-51"></a>    <span class='hs-varid'>a</span><span class='hs-conop'>:</span><span class='hs-keyword'>as</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varop'>$</span> <span class='hs-varid'>es</span> <span class='hs-varop'>++</span> <span class='hs-varid'>map</span> <span class='hs-varid'>reverse</span> <span class='hs-varid'>es</span>
<a name="line-52"></a>    <span class='hs-varid'>auts</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts</span> <span class='hs-varid'>g</span>
<a name="line-53"></a>
<a name="line-54"></a><a name="isArcTransitive'"></a><span class='hs-definition'>isArcTransitive'</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-55"></a>    <span class='hs-varid'>orbitP</span> <span class='hs-varid'>auts</span> <span class='hs-varid'>v</span> <span class='hs-varop'>==</span> <span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-comment'>-- isVertexTransitive g</span>
<a name="line-56"></a>    <span class='hs-varid'>orbitP</span> <span class='hs-varid'>stab</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-varid'>n</span><span class='hs-conop'>:</span><span class='hs-varid'>ns</span>
<a name="line-57"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>auts</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts</span> <span class='hs-varid'>g</span>
<a name="line-58"></a>          <span class='hs-varid'>stab</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>dropWhile</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>v</span> <span class='hs-varop'>.^</span> <span class='hs-varid'>p</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varid'>auts</span> <span class='hs-comment'>-- we know that graphAuts are returned in this order</span>
<a name="line-59"></a>          <span class='hs-varid'>n</span><span class='hs-conop'>:</span><span class='hs-varid'>ns</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>nbrs</span> <span class='hs-varid'>g</span> <span class='hs-varid'>v</span>
<a name="line-60"></a>
<a name="line-61"></a><span class='hs-comment'>-- execution time of both of the above is dominated by the time to calculate the graph auts, so their performance is similar</span>
<a name="line-62"></a>
<a name="line-63"></a>
<a name="line-64"></a><span class='hs-comment'>-- then k n, kb n n, q n, other platonic solids, petersen graph, heawood graph, pappus graph, desargues graph are all arc-transitive</span>
<a name="line-65"></a>
<a name="line-66"></a>
<a name="line-67"></a><a name="findArcs"></a><span class='hs-comment'>-- find arcs of length l from x using dfs - results returned in order</span>
<a name="line-68"></a><span class='hs-comment'>-- an arc is a sequence of vertices connected by edges, no doubling back, but self-crossings allowed</span>
<a name="line-69"></a><span class='hs-definition'>findArcs</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-varid'>l</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-varid'>reverse</span> <span class='hs-varop'>$</span> <span class='hs-varid'>dfs</span> <span class='hs-keyglyph'>[</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>x</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>]</span> <span class='hs-keyword'>where</span>
<a name="line-70"></a>    <span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span> <span class='hs-layout'>(</span><span class='hs-varid'>z1</span><span class='hs-conop'>:</span><span class='hs-varid'>z2</span><span class='hs-conop'>:</span><span class='hs-varid'>zs</span><span class='hs-layout'>,</span><span class='hs-varid'>l'</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>nodes</span><span class='hs-layout'>)</span>
<a name="line-71"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>l</span> <span class='hs-varop'>==</span> <span class='hs-varid'>l'</span>   <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>z1</span><span class='hs-conop'>:</span><span class='hs-varid'>z2</span><span class='hs-conop'>:</span><span class='hs-varid'>zs</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>dfs</span> <span class='hs-varid'>nodes</span>
<a name="line-72"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>dfs</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>w</span><span class='hs-conop'>:</span><span class='hs-varid'>z1</span><span class='hs-conop'>:</span><span class='hs-varid'>z2</span><span class='hs-conop'>:</span><span class='hs-varid'>zs</span><span class='hs-layout'>,</span><span class='hs-varid'>l'</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>w</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>nbrs</span> <span class='hs-varid'>g</span> <span class='hs-varid'>z1</span><span class='hs-layout'>,</span> <span class='hs-varid'>w</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>z2</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>++</span> <span class='hs-varid'>nodes</span>
<a name="line-73"></a>    <span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>z</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span><span class='hs-varid'>l'</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>nodes</span><span class='hs-layout'>)</span>
<a name="line-74"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>l</span> <span class='hs-varop'>==</span> <span class='hs-varid'>l'</span>   <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>z</span><span class='hs-keyglyph'>]</span> <span class='hs-conop'>:</span> <span class='hs-varid'>dfs</span> <span class='hs-varid'>nodes</span>
<a name="line-75"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>dfs</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>w</span><span class='hs-layout'>,</span><span class='hs-varid'>z</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span><span class='hs-varid'>l'</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>w</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>nbrs</span> <span class='hs-varid'>g</span> <span class='hs-varid'>z</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>++</span> <span class='hs-varid'>nodes</span>
<a name="line-76"></a>    <span class='hs-varid'>dfs</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-77"></a>
<a name="line-78"></a><span class='hs-comment'>-- note that a graph with triangles can't be 3-arc transitive, etc, because an aut can't map a self-crossing arc to a non-self-crossing arc</span>
<a name="line-79"></a>
<a name="line-80"></a><a name="isnArcTransitive"></a><span class='hs-comment'>-- |A graph is n-arc-transitive is its automorphism group is transitive on n-arcs. (An n-arc is an ordered sequence (v0,...,vn) of adjacent vertices, with crossings allowed but not doubling back.)</span>
<a name="line-81"></a><span class='hs-definition'>isnArcTransitive</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-82"></a><span class='hs-definition'>isnArcTransitive</span> <span class='hs-keyword'>_</span> <span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span>
<a name="line-83"></a><span class='hs-definition'>isnArcTransitive</span> <span class='hs-varid'>n</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-84"></a>    <span class='hs-varid'>orbitP</span> <span class='hs-varid'>auts</span> <span class='hs-varid'>v</span> <span class='hs-varop'>==</span> <span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-comment'>-- isVertexTransitive g</span>
<a name="line-85"></a>    <span class='hs-varid'>orbit</span> <span class='hs-layout'>(</span><span class='hs-varop'>-&gt;^</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span> <span class='hs-varid'>stab</span> <span class='hs-varop'>==</span> <span class='hs-varid'>a</span><span class='hs-conop'>:</span><span class='hs-keyword'>as</span>
<a name="line-86"></a>    <span class='hs-comment'>-- closure [a] [ -&gt;^ h | h &lt;- stab] == a:as</span>
<a name="line-87"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>auts</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts</span> <span class='hs-varid'>g</span>
<a name="line-88"></a>          <span class='hs-varid'>stab</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>dropWhile</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>v</span> <span class='hs-varop'>.^</span> <span class='hs-varid'>p</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varid'>auts</span> <span class='hs-comment'>-- we know that graphAuts are returned in this order</span>
<a name="line-89"></a>          <span class='hs-varid'>a</span><span class='hs-conop'>:</span><span class='hs-keyword'>as</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>findArcs</span> <span class='hs-varid'>g</span> <span class='hs-varid'>v</span> <span class='hs-varid'>n</span>
<a name="line-90"></a>
<a name="line-91"></a><a name="is2ArcTransitive"></a><span class='hs-definition'>is2ArcTransitive</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-92"></a><span class='hs-definition'>is2ArcTransitive</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>isnArcTransitive</span> <span class='hs-num'>2</span> <span class='hs-varid'>g</span>
<a name="line-93"></a>
<a name="line-94"></a><a name="is3ArcTransitive"></a><span class='hs-definition'>is3ArcTransitive</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-95"></a><span class='hs-definition'>is3ArcTransitive</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>isnArcTransitive</span> <span class='hs-num'>3</span> <span class='hs-varid'>g</span>
<a name="line-96"></a>
<a name="line-97"></a><a name="isDistanceTransitive"></a><span class='hs-comment'>-- Godsil &amp; Royle 66-7</span>
<a name="line-98"></a><span class='hs-comment'>-- |A graph is distance transitive if given any two ordered pairs of vertices (u,u') and (v,v') with d(u,u') == d(v,v'),</span>
<a name="line-99"></a><span class='hs-comment'>-- there is an automorphism of the graph that takes (u,u') to (v,v')</span>
<a name="line-100"></a><span class='hs-definition'>isDistanceTransitive</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-101"></a><span class='hs-definition'>isDistanceTransitive</span> <span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span>
<a name="line-102"></a><span class='hs-definition'>isDistanceTransitive</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span>
<a name="line-103"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>isConnected</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span>
<a name="line-104"></a>        <span class='hs-varid'>orbitP</span> <span class='hs-varid'>auts</span> <span class='hs-varid'>v</span> <span class='hs-varop'>==</span> <span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-comment'>-- isVertexTransitive g</span>
<a name="line-105"></a>        <span class='hs-varid'>length</span> <span class='hs-varid'>stabOrbits</span> <span class='hs-varop'>==</span> <span class='hs-varid'>diameter</span> <span class='hs-varid'>g</span> <span class='hs-varop'>+</span> <span class='hs-num'>1</span> <span class='hs-comment'>-- the orbits under the stabiliser of v coincide with the distance partition from v</span>
<a name="line-106"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"isDistanceTransitive: only defined for connected graphs"</span>
<a name="line-107"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>auts</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts</span> <span class='hs-varid'>g</span>
<a name="line-108"></a>          <span class='hs-varid'>stab</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>dropWhile</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>v</span> <span class='hs-varop'>.^</span> <span class='hs-varid'>p</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varid'>auts</span> <span class='hs-comment'>-- we know that graphAuts are returned in this order</span>
<a name="line-109"></a>          <span class='hs-varid'>stabOrbits</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>os</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>orbits</span> <span class='hs-varid'>stab</span> <span class='hs-keyword'>in</span> <span class='hs-varid'>os</span> <span class='hs-varop'>++</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-conop'>:</span><span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-conid'>L</span><span class='hs-varop'>.\\</span> <span class='hs-varid'>concat</span> <span class='hs-varid'>os</span><span class='hs-layout'>)</span> <span class='hs-comment'>-- include fixed point orbits</span>
<a name="line-110"></a>
<a name="line-111"></a>
<a name="line-112"></a><span class='hs-comment'>-- GRAPH AUTOMORPHISMS</span>
<a name="line-113"></a>
<a name="line-114"></a><span class='hs-comment'>-- !! Note, in the literature the following is just called the intersection of two partitions</span>
<a name="line-115"></a><span class='hs-comment'>-- !! Refinement actually refers to the process of refining to an equitable partition</span>
<a name="line-116"></a>
<a name="line-117"></a><a name="refine"></a><span class='hs-comment'>-- refine one partition by another</span>
<a name="line-118"></a><span class='hs-definition'>refine</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>refine'</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span>
<a name="line-119"></a><span class='hs-comment'>-- Refinement preserves ordering within cells but not between cells</span>
<a name="line-120"></a><span class='hs-comment'>-- eg the cell [1,2,3,4] could be refined to [2,4],[1,3]</span>
<a name="line-121"></a>
<a name="line-122"></a><a name="refine'"></a><span class='hs-comment'>-- refine, but leaving null cells in</span>
<a name="line-123"></a><span class='hs-comment'>-- we use this in the graphAuts functions when comparing two refinements to check that they split in the same way</span>
<a name="line-124"></a><span class='hs-definition'>refine'</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>c1</span> <span class='hs-varop'>`intersect`</span> <span class='hs-varid'>c2</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>c2</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>p2</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>c1</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>p1</span><span class='hs-keyglyph'>]</span>
<a name="line-125"></a>
<a name="line-126"></a>
<a name="line-127"></a><a name="isGraphAut"></a><span class='hs-definition'>isGraphAut</span> <span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-varid'>h</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>all</span> <span class='hs-layout'>(</span><span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>e</span> <span class='hs-varop'>-^</span> <span class='hs-varid'>h</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>es</span><span class='hs-keyglyph'>]</span>
<a name="line-128"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>es'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varid'>es</span>
<a name="line-129"></a><span class='hs-comment'>-- this works best on sparse graphs, where p(edge) &lt; 1/2</span>
<a name="line-130"></a><span class='hs-comment'>-- if p(edge) &gt; 1/2, it would be better to test on the complement of the graph</span>
<a name="line-131"></a>
<a name="line-132"></a>
<a name="line-133"></a>
<a name="line-134"></a>
<a name="line-135"></a><a name="adjLists"></a><span class='hs-comment'>-- Calculate a map consisting of neighbour lists for each vertex in the graph</span>
<a name="line-136"></a><span class='hs-comment'>-- If a vertex has no neighbours then it is left out of the map</span>
<a name="line-137"></a><span class='hs-definition'>adjLists</span> <span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>adjLists'</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>empty</span> <span class='hs-varid'>es</span>
<a name="line-138"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>adjLists'</span> <span class='hs-varid'>nbrs</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>u</span><span class='hs-layout'>,</span><span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span><span class='hs-conop'>:</span><span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-139"></a>              <span class='hs-varid'>adjLists'</span> <span class='hs-layout'>(</span><span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>insertWith'</span> <span class='hs-layout'>(</span><span class='hs-varid'>flip</span> <span class='hs-layout'>(</span><span class='hs-varop'>++</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>u</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>$</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>insertWith'</span> <span class='hs-layout'>(</span><span class='hs-varid'>flip</span> <span class='hs-layout'>(</span><span class='hs-varop'>++</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>u</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span> <span class='hs-varid'>nbrs</span><span class='hs-layout'>)</span> <span class='hs-varid'>es</span>
<a name="line-140"></a>          <span class='hs-varid'>adjLists'</span> <span class='hs-varid'>nbrs</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>nbrs</span>
<a name="line-141"></a>
<a name="line-142"></a>
<a name="line-143"></a><span class='hs-comment'>-- ALTERNATIVE VERSIONS OF GRAPH AUTS</span>
<a name="line-144"></a><span class='hs-comment'>-- (showing how we got to the final version)</span>
<a name="line-145"></a>
<a name="line-146"></a><a name="graphAuts1"></a><span class='hs-comment'>-- return all graph automorphisms, using naive depth first search</span>
<a name="line-147"></a><span class='hs-definition'>graphAuts1</span> <span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>dfs</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>vs</span>
<a name="line-148"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>dfs</span> <span class='hs-varid'>xys</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varid'>ys</span> <span class='hs-keyglyph'>=</span>
<a name="line-149"></a>              <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>xys</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>ys</span><span class='hs-layout'>,</span> <span class='hs-varid'>isCompatible</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varid'>xys</span><span class='hs-keyglyph'>]</span>
<a name="line-150"></a>          <span class='hs-varid'>dfs</span> <span class='hs-varid'>xys</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>fromPairs</span> <span class='hs-varid'>xys</span><span class='hs-keyglyph'>]</span>
<a name="line-151"></a>          <span class='hs-varid'>isCompatible</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varid'>xys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>and</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>x'</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-keyglyph'>]</span>
<a name="line-152"></a>          <span class='hs-varid'>es'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varid'>es</span>
<a name="line-153"></a>
<a name="line-154"></a><a name="graphAuts2"></a><span class='hs-comment'>-- return generators for graph automorphisms</span>
<a name="line-155"></a><span class='hs-comment'>-- (using Lemma 9.1.1 from Seress p203 to prune the search tree)</span>
<a name="line-156"></a><span class='hs-definition'>graphAuts2</span> <span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>vs</span>
<a name="line-157"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-varid'>us</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-158"></a>              <span class='hs-keyword'>let</span> <span class='hs-varid'>uus</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zip</span> <span class='hs-varid'>us</span> <span class='hs-varid'>us</span>
<a name="line-159"></a>              <span class='hs-keyword'>in</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>take</span> <span class='hs-num'>1</span> <span class='hs-varop'>$</span> <span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span><span class='hs-varid'>w</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>uus</span><span class='hs-layout'>)</span> <span class='hs-varid'>vs</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span> <span class='hs-conop'>:</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>w</span> <span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>w</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vs</span><span class='hs-layout'>,</span> <span class='hs-varid'>isCompatible</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span><span class='hs-varid'>w</span><span class='hs-layout'>)</span> <span class='hs-varid'>uus</span><span class='hs-keyglyph'>]</span>
<a name="line-160"></a>              <span class='hs-varop'>++</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>us</span><span class='hs-layout'>)</span> <span class='hs-varid'>vs</span>
<a name="line-161"></a>              <span class='hs-comment'>-- stab us == transversal for stab (v:us) ++ stab (v:us)  (generators thereof)</span>
<a name="line-162"></a>          <span class='hs-varid'>graphAuts'</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span> <span class='hs-comment'>-- we're not interested in finding the identity element</span>
<a name="line-163"></a>          <span class='hs-varid'>dfs</span> <span class='hs-varid'>xys</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varid'>ys</span> <span class='hs-keyglyph'>=</span>
<a name="line-164"></a>              <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>xys</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>ys</span><span class='hs-layout'>,</span> <span class='hs-varid'>isCompatible</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varid'>xys</span><span class='hs-keyglyph'>]</span>
<a name="line-165"></a>          <span class='hs-varid'>dfs</span> <span class='hs-varid'>xys</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>fromPairs</span> <span class='hs-varid'>xys</span><span class='hs-keyglyph'>]</span>
<a name="line-166"></a>          <span class='hs-varid'>isCompatible</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varid'>xys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>and</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>x'</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-keyglyph'>]</span>
<a name="line-167"></a>          <span class='hs-varid'>es'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varid'>es</span>
<a name="line-168"></a>
<a name="line-169"></a><a name="graphAuts3"></a><span class='hs-comment'>-- Now using distance partitions</span>
<a name="line-170"></a><span class='hs-comment'>-- Note that because of the use of distance partitions, this is only valid for connected graphs</span>
<a name="line-171"></a><span class='hs-definition'>graphAuts3</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>vs</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>where</span>
<a name="line-172"></a>    <span class='hs-varid'>graphAuts'</span> <span class='hs-varid'>us</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-173"></a>        <span class='hs-keyword'>let</span> <span class='hs-varid'>px</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-varid'>ys</span> <span class='hs-conop'>:</span> <span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-174"></a>            <span class='hs-varid'>p</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span>
<a name="line-175"></a>            <span class='hs-varid'>uus</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zip</span> <span class='hs-varid'>us</span> <span class='hs-varid'>us</span>
<a name="line-176"></a>            <span class='hs-varid'>p'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varop'>$</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>px</span>
<a name="line-177"></a>        <span class='hs-keyword'>in</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>take</span> <span class='hs-num'>1</span> <span class='hs-varop'>$</span> <span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>uus</span><span class='hs-layout'>)</span> <span class='hs-varid'>px</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-178"></a>        <span class='hs-varop'>++</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>us</span><span class='hs-layout'>)</span> <span class='hs-varid'>p'</span>
<a name="line-179"></a>    <span class='hs-varid'>graphAuts'</span> <span class='hs-varid'>us</span> <span class='hs-layout'>(</span><span class='hs-conid'>[]</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-varid'>us</span> <span class='hs-varid'>pt</span>
<a name="line-180"></a>    <span class='hs-varid'>graphAuts'</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-181"></a>    <span class='hs-varid'>dfs</span> <span class='hs-varid'>xys</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span>
<a name="line-182"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>p1</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>p2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-183"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span>
<a name="line-184"></a>            <span class='hs-keyword'>let</span> <span class='hs-varid'>p1'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varid'>p1</span>
<a name="line-185"></a>                <span class='hs-varid'>p2'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varid'>p2</span>
<a name="line-186"></a>            <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>all</span> <span class='hs-varid'>isSingleton</span> <span class='hs-varid'>p1'</span>
<a name="line-187"></a>               <span class='hs-keyword'>then</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>xys'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>xys</span> <span class='hs-varop'>++</span> <span class='hs-varid'>zip</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>p1'</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>p2'</span><span class='hs-layout'>)</span>
<a name="line-188"></a>                    <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>isCompatible</span> <span class='hs-varid'>xys'</span> <span class='hs-keyword'>then</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>fromPairs'</span> <span class='hs-varid'>xys'</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>else</span> <span class='hs-conid'>[]</span>
<a name="line-189"></a>                    <span class='hs-comment'>-- we shortcut the search when we have all singletons, so must check isCompatible to avoid false positives</span>
<a name="line-190"></a>               <span class='hs-keyword'>else</span> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>p1''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p1'</span>
<a name="line-191"></a>                        <span class='hs-varid'>ys</span><span class='hs-conop'>:</span><span class='hs-varid'>p2''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p2'</span>
<a name="line-192"></a>                    <span class='hs-keyword'>in</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>xys</span><span class='hs-layout'>)</span>
<a name="line-193"></a>                                   <span class='hs-layout'>(</span><span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-varid'>xs</span> <span class='hs-conop'>:</span> <span class='hs-varid'>p1''</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-194"></a>                                   <span class='hs-layout'>(</span><span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>p2''</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-195"></a>                                   <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-196"></a>    <span class='hs-varid'>isCompatible</span> <span class='hs-varid'>xys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>and</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>x'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span> <span class='hs-varop'>&lt;</span> <span class='hs-varid'>x'</span><span class='hs-keyglyph'>]</span>
<a name="line-197"></a>    <span class='hs-varid'>dps</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vs</span><span class='hs-keyglyph'>]</span>
<a name="line-198"></a>    <span class='hs-varid'>es'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varid'>es</span>
<a name="line-199"></a>
<a name="line-200"></a><a name="isSingleton"></a><span class='hs-definition'>isSingleton</span> <span class='hs-keyglyph'>[</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span>
<a name="line-201"></a><span class='hs-definition'>isSingleton</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-202"></a>
<a name="line-203"></a>
<a name="line-204"></a><a name="graphAuts4"></a><span class='hs-comment'>-- Now we try to use generators we've already found at a given level to save us having to look for others</span>
<a name="line-205"></a><span class='hs-comment'>-- For example, if we have found (1 2)(3 4) and (1 3 2), then we don't need to look for something taking 1 -&gt; 4</span>
<a name="line-206"></a><span class='hs-definition'>graphAuts4</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>vs</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>where</span>
<a name="line-207"></a>    <span class='hs-varid'>graphAuts'</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-208"></a>        <span class='hs-comment'>-- let p' = L.sort $ filter (not . null) $ refine' (ys:pt) (dps M.! x)</span>
<a name="line-209"></a>        <span class='hs-keyword'>let</span> <span class='hs-varid'>p'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varop'>$</span> <span class='hs-varid'>refine</span> <span class='hs-layout'>(</span><span class='hs-varid'>ys</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-210"></a>        <span class='hs-keyword'>in</span> <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span> <span class='hs-varid'>x</span> <span class='hs-varid'>ys</span> <span class='hs-conid'>[]</span>
<a name="line-211"></a>        <span class='hs-varop'>++</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>us</span><span class='hs-layout'>)</span> <span class='hs-varid'>p'</span>
<a name="line-212"></a>    <span class='hs-varid'>graphAuts'</span> <span class='hs-varid'>us</span> <span class='hs-layout'>(</span><span class='hs-conid'>[]</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-varid'>us</span> <span class='hs-varid'>pt</span>
<a name="line-213"></a>    <span class='hs-varid'>graphAuts'</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-214"></a>    <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>ph</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-varid'>hs</span> <span class='hs-keyglyph'>=</span>
<a name="line-215"></a>        <span class='hs-keyword'>let</span> <span class='hs-varid'>px</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>x</span> <span class='hs-varid'>ph</span> <span class='hs-conop'>:</span> <span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-216"></a>            <span class='hs-varid'>py</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ph</span> <span class='hs-conop'>:</span> <span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span>
<a name="line-217"></a>            <span class='hs-varid'>uus</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zip</span> <span class='hs-varid'>us</span> <span class='hs-varid'>us</span>
<a name="line-218"></a>        <span class='hs-keyword'>in</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>uus</span><span class='hs-layout'>)</span> <span class='hs-varid'>px</span> <span class='hs-varid'>py</span> <span class='hs-keyword'>of</span>
<a name="line-219"></a>           <span class='hs-conid'>[]</span>  <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span> <span class='hs-varid'>x</span> <span class='hs-varid'>ys</span> <span class='hs-varid'>hs</span>
<a name="line-220"></a>           <span class='hs-varid'>h</span><span class='hs-conop'>:</span><span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>hs'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>h</span><span class='hs-conop'>:</span><span class='hs-varid'>hs</span> <span class='hs-keyword'>in</span> <span class='hs-varid'>h</span> <span class='hs-conop'>:</span> <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>ys</span> <span class='hs-conid'>L</span><span class='hs-varop'>.\\</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-varop'>.^^</span> <span class='hs-varid'>hs'</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>hs'</span>
<a name="line-221"></a>    <span class='hs-varid'>level</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-222"></a>    <span class='hs-varid'>dfs</span> <span class='hs-varid'>xys</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span>
<a name="line-223"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>p1</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>p2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-224"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span>
<a name="line-225"></a>            <span class='hs-keyword'>let</span> <span class='hs-varid'>p1'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varid'>p1</span>
<a name="line-226"></a>                <span class='hs-varid'>p2'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varid'>p2</span>
<a name="line-227"></a>            <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>all</span> <span class='hs-varid'>isSingleton</span> <span class='hs-varid'>p1'</span>
<a name="line-228"></a>               <span class='hs-keyword'>then</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>xys'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>xys</span> <span class='hs-varop'>++</span> <span class='hs-varid'>zip</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>p1'</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>p2'</span><span class='hs-layout'>)</span>
<a name="line-229"></a>                    <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>isCompatible</span> <span class='hs-varid'>xys'</span> <span class='hs-keyword'>then</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>fromPairs'</span> <span class='hs-varid'>xys'</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>else</span> <span class='hs-conid'>[]</span>
<a name="line-230"></a>               <span class='hs-keyword'>else</span> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>p1''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p1'</span>
<a name="line-231"></a>                        <span class='hs-varid'>ys</span><span class='hs-conop'>:</span><span class='hs-varid'>p2''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p2'</span>
<a name="line-232"></a>                    <span class='hs-keyword'>in</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>xys</span><span class='hs-layout'>)</span>
<a name="line-233"></a>                                   <span class='hs-layout'>(</span><span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-varid'>xs</span> <span class='hs-conop'>:</span> <span class='hs-varid'>p1''</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-234"></a>                                   <span class='hs-layout'>(</span><span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>p2''</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-235"></a>                                   <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-236"></a>    <span class='hs-varid'>isCompatible</span> <span class='hs-varid'>xys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>and</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>x'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span> <span class='hs-varop'>&lt;</span> <span class='hs-varid'>x'</span><span class='hs-keyglyph'>]</span>
<a name="line-237"></a>    <span class='hs-varid'>dps</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vs</span><span class='hs-keyglyph'>]</span>
<a name="line-238"></a>    <span class='hs-varid'>es'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varid'>es</span>
<a name="line-239"></a>
<a name="line-240"></a><span class='hs-comment'>-- contrary to first thought, you can't stop when a level is null - eg kb 2 3, the third level is null, but the fourth isn't</span>
<a name="line-241"></a>
<a name="line-242"></a>
<a name="line-243"></a>
<a name="line-244"></a><a name="eqgraph"></a><span class='hs-comment'>-- an example for equitable partitions</span>
<a name="line-245"></a><span class='hs-comment'>-- this is a graph whose distance partition (from any vertex) can be refined to an equitable partition</span>
<a name="line-246"></a><span class='hs-definition'>eqgraph</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span> <span class='hs-keyword'>where</span>
<a name="line-247"></a>    <span class='hs-varid'>vs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-num'>14</span><span class='hs-keyglyph'>]</span>
<a name="line-248"></a>    <span class='hs-varid'>es</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>[</span><span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-layout'>,</span><span class='hs-num'>14</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span><span class='hs-keyglyph'>[</span><span class='hs-num'>2</span><span class='hs-layout'>,</span><span class='hs-num'>13</span><span class='hs-keyglyph'>]</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>v1</span><span class='hs-layout'>,</span><span class='hs-varid'>v2</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>|</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>v1</span><span class='hs-layout'>,</span><span class='hs-varid'>v2</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>combinationsOf</span> <span class='hs-num'>2</span> <span class='hs-varid'>vs</span><span class='hs-layout'>,</span> <span class='hs-varid'>v1</span><span class='hs-varop'>+</span><span class='hs-num'>1</span> <span class='hs-varop'>==</span> <span class='hs-varid'>v2</span> <span class='hs-varop'>||</span> <span class='hs-varid'>v1</span><span class='hs-varop'>+</span><span class='hs-num'>3</span> <span class='hs-varop'>==</span> <span class='hs-varid'>v2</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>even</span> <span class='hs-varid'>v2</span><span class='hs-keyglyph'>]</span>
<a name="line-249"></a>
<a name="line-250"></a><a name="toEquitable"></a><span class='hs-comment'>-- refine a partition to give an equitable partition</span>
<a name="line-251"></a><span class='hs-definition'>toEquitable</span> <span class='hs-varid'>g</span> <span class='hs-varid'>cells</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varop'>$</span> <span class='hs-varid'>toEquitable'</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>cells</span> <span class='hs-keyword'>where</span>
<a name="line-252"></a>    <span class='hs-varid'>toEquitable'</span> <span class='hs-varid'>ls</span> <span class='hs-layout'>(</span><span class='hs-varid'>r</span><span class='hs-conop'>:</span><span class='hs-varid'>rs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-253"></a>        <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>lls</span><span class='hs-layout'>,</span><span class='hs-varid'>lrs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>partition</span> <span class='hs-varid'>isSingleton</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>splitNumNbrs</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-varid'>ls</span>
<a name="line-254"></a>            <span class='hs-comment'>-- so the lrs split, and the lls didn't</span>
<a name="line-255"></a>            <span class='hs-varid'>rs'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>concatMap</span> <span class='hs-layout'>(</span><span class='hs-varid'>splitNumNbrs</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-varid'>rs</span>
<a name="line-256"></a>        <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>isSingleton</span> <span class='hs-varid'>r</span> <span class='hs-comment'>-- then we know it won't split further, so can remove it from further processing</span>
<a name="line-257"></a>           <span class='hs-keyword'>then</span> <span class='hs-varid'>r</span> <span class='hs-conop'>:</span> <span class='hs-varid'>toEquitable'</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>lls</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>lrs</span> <span class='hs-varop'>++</span> <span class='hs-varid'>rs'</span><span class='hs-layout'>)</span>
<a name="line-258"></a>           <span class='hs-keyword'>else</span> <span class='hs-varid'>toEquitable'</span> <span class='hs-layout'>(</span><span class='hs-varid'>r</span> <span class='hs-conop'>:</span> <span class='hs-varid'>concat</span> <span class='hs-varid'>lls</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>lrs</span> <span class='hs-varop'>++</span> <span class='hs-varid'>rs'</span><span class='hs-layout'>)</span>
<a name="line-259"></a>    <span class='hs-varid'>toEquitable'</span> <span class='hs-varid'>ls</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ls</span>
<a name="line-260"></a>    <span class='hs-varid'>splitNumNbrs</span> <span class='hs-varid'>t</span> <span class='hs-varid'>c</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-varid'>snd</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>groupBy</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>fst</span> <span class='hs-varid'>x</span> <span class='hs-varop'>==</span> <span class='hs-varid'>fst</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span>
<a name="line-261"></a>                    <span class='hs-keyglyph'>[</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>nbrs_g</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varop'>`intersect`</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>c</span><span class='hs-keyglyph'>]</span>
<a name="line-262"></a>    <span class='hs-varid'>nbrs_g</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>nbrs</span> <span class='hs-varid'>g</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vertices</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>]</span>
<a name="line-263"></a>
<a name="line-264"></a>
<a name="line-265"></a><a name="toEquitable2"></a><span class='hs-comment'>-- try to refine two partitions in parallel, failing if they become mismatched</span>
<a name="line-266"></a><span class='hs-definition'>toEquitable2</span> <span class='hs-varid'>nbrs_g</span> <span class='hs-varid'>psrc</span> <span class='hs-varid'>ptrg</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unzip</span> <span class='hs-varop'>$</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varop'>$</span> <span class='hs-varid'>toEquitable'</span> <span class='hs-conid'>[]</span> <span class='hs-layout'>(</span><span class='hs-varid'>zip</span> <span class='hs-varid'>psrc</span> <span class='hs-varid'>ptrg</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-267"></a>    <span class='hs-varid'>toEquitable'</span> <span class='hs-varid'>ls</span> <span class='hs-layout'>(</span><span class='hs-varid'>r</span><span class='hs-conop'>:</span><span class='hs-varid'>rs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-268"></a>        <span class='hs-keyword'>let</span> <span class='hs-varid'>ls'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>splitNumNbrs</span> <span class='hs-varid'>nbrs_g</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-varid'>ls</span>
<a name="line-269"></a>            <span class='hs-layout'>(</span><span class='hs-varid'>lls</span><span class='hs-layout'>,</span><span class='hs-varid'>lrs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>partition</span> <span class='hs-varid'>isSingleton</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map</span> <span class='hs-varid'>fromJust</span> <span class='hs-varid'>ls'</span>
<a name="line-270"></a>            <span class='hs-varid'>rs'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>splitNumNbrs</span> <span class='hs-varid'>nbrs_g</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-varid'>rs</span>
<a name="line-271"></a>        <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>any</span> <span class='hs-varid'>isNothing</span> <span class='hs-varid'>ls'</span> <span class='hs-varop'>||</span> <span class='hs-varid'>any</span> <span class='hs-varid'>isNothing</span> <span class='hs-varid'>rs'</span>
<a name="line-272"></a>           <span class='hs-keyword'>then</span> <span class='hs-conid'>[]</span>
<a name="line-273"></a>           <span class='hs-keyword'>else</span>
<a name="line-274"></a>               <span class='hs-comment'>{- if (isSingleton . fst) r
<a name="line-275"></a>               then r : toEquitable' (concat lls) (concat lrs ++ concatMap fromJust rs')
<a name="line-276"></a>               else -}</span> <span class='hs-varid'>toEquitable'</span> <span class='hs-layout'>(</span><span class='hs-varid'>r</span> <span class='hs-conop'>:</span> <span class='hs-varid'>concat</span> <span class='hs-varid'>lls</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>lrs</span> <span class='hs-varop'>++</span> <span class='hs-varid'>concatMap</span> <span class='hs-varid'>fromJust</span> <span class='hs-varid'>rs'</span><span class='hs-layout'>)</span>
<a name="line-277"></a>    <span class='hs-varid'>toEquitable'</span> <span class='hs-varid'>ls</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ls</span>
<a name="line-278"></a>
<a name="line-279"></a><a name="splitNumNbrs"></a><span class='hs-definition'>splitNumNbrs</span> <span class='hs-varid'>nbrs_g</span> <span class='hs-layout'>(</span><span class='hs-varid'>t_src</span><span class='hs-layout'>,</span><span class='hs-varid'>t_trg</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>c_src</span><span class='hs-layout'>,</span><span class='hs-varid'>c_trg</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-280"></a>    <span class='hs-keyword'>let</span> <span class='hs-varid'>src_split</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>groupBy</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>fst</span> <span class='hs-varid'>x</span> <span class='hs-varop'>==</span> <span class='hs-varid'>fst</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span>
<a name="line-281"></a>                    <span class='hs-keyglyph'>[</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>nbrs_g</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varop'>`intersect`</span> <span class='hs-varid'>t_src</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>c_src</span><span class='hs-keyglyph'>]</span>
<a name="line-282"></a>        <span class='hs-varid'>trg_split</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>groupBy</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>fst</span> <span class='hs-varid'>x</span> <span class='hs-varop'>==</span> <span class='hs-varid'>fst</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span>
<a name="line-283"></a>                    <span class='hs-keyglyph'>[</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>nbrs_g</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varop'>`intersect`</span> <span class='hs-varid'>t_trg</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>c_trg</span><span class='hs-keyglyph'>]</span>
<a name="line-284"></a>    <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>src_split</span> <span class='hs-varop'>==</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>trg_split</span>
<a name="line-285"></a>       <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>fst</span> <span class='hs-varop'>.</span> <span class='hs-varid'>head</span><span class='hs-layout'>)</span> <span class='hs-varid'>src_split</span> <span class='hs-varop'>==</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>fst</span> <span class='hs-varop'>.</span> <span class='hs-varid'>head</span><span class='hs-layout'>)</span> <span class='hs-varid'>trg_split</span>
<a name="line-286"></a>       <span class='hs-keyword'>then</span> <span class='hs-conid'>Just</span> <span class='hs-varop'>$</span> <span class='hs-varid'>zip</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-varid'>snd</span><span class='hs-layout'>)</span> <span class='hs-varid'>src_split</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-varid'>snd</span><span class='hs-layout'>)</span> <span class='hs-varid'>trg_split</span><span class='hs-layout'>)</span>
<a name="line-287"></a>       <span class='hs-keyword'>else</span> <span class='hs-conid'>Nothing</span>
<a name="line-288"></a>       <span class='hs-comment'>-- else error (show (src_split, trg_split)) -- for debugging</span>
<a name="line-289"></a>
<a name="line-290"></a><span class='hs-comment'>-- Now, every time we intersect two partitions, refine to an equitable partition</span>
<a name="line-291"></a>
<a name="line-292"></a><a name="graphAuts"></a><span class='hs-comment'>-- |Given a graph g, @graphAuts g@ returns generators for the automorphism group of g.</span>
<a name="line-293"></a><span class='hs-comment'>-- If g is connected, then the generators will be a strong generating set.</span>
<a name="line-294"></a><span class='hs-definition'>graphAuts</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Permutation</span> <span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span>
<a name="line-295"></a><span class='hs-definition'>graphAuts</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>autsWithinComponents</span> <span class='hs-varop'>++</span> <span class='hs-varid'>isosBetweenComponents</span>
<a name="line-296"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>cs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>inducedSubgraph</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>components</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span>
<a name="line-297"></a>          <span class='hs-comment'>-- autsWithinComponents = concatMap graphAutsCon cs</span>
<a name="line-298"></a>          <span class='hs-varid'>autsWithinComponents</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>concatMap</span> <span class='hs-varid'>graphAuts4</span> <span class='hs-varid'>cs</span>
<a name="line-299"></a>          <span class='hs-varid'>isosBetweenComponents</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-varid'>swapFromIso</span> <span class='hs-varop'>$</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>take</span> <span class='hs-num'>1</span> <span class='hs-layout'>(</span><span class='hs-varid'>graphIsos</span> <span class='hs-varid'>ci</span> <span class='hs-varid'>cj</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>ci</span><span class='hs-layout'>,</span><span class='hs-varid'>cj</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>pairs</span> <span class='hs-varid'>cs</span><span class='hs-keyglyph'>]</span>
<a name="line-300"></a>          <span class='hs-varid'>swapFromIso</span> <span class='hs-varid'>xys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromPairs</span> <span class='hs-layout'>(</span><span class='hs-varid'>xys</span> <span class='hs-varop'>++</span> <span class='hs-varid'>map</span> <span class='hs-varid'>swap</span> <span class='hs-varid'>xys</span><span class='hs-layout'>)</span>
<a name="line-301"></a>          <span class='hs-varid'>swap</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-302"></a><span class='hs-comment'>-- Using graphAuts4 instead of graphAutsCon as latter appears to have a bug, eg</span>
<a name="line-303"></a><span class='hs-comment'>-- &gt; graphAuts4 $ G [1..3] [[1,2],[2,3]]</span>
<a name="line-304"></a><span class='hs-comment'>-- [[[1,3]]]</span>
<a name="line-305"></a><span class='hs-comment'>-- &gt; graphAutsCon $ G [1..3] [[1,2],[2,3]]</span>
<a name="line-306"></a><span class='hs-comment'>-- []</span>
<a name="line-307"></a>
<a name="line-308"></a><a name="graphAutsCon"></a><span class='hs-comment'>-- Automorphisms of a connected graph</span>
<a name="line-309"></a><span class='hs-definition'>graphAutsCon</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span>
<a name="line-310"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>isConnected</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-conid'>[]</span> <span class='hs-layout'>(</span><span class='hs-varid'>toEquitable</span> <span class='hs-varid'>g</span> <span class='hs-varop'>$</span> <span class='hs-varid'>valencyPartition</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span>
<a name="line-311"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"graphAutsCon: graph is not connected"</span>
<a name="line-312"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-313"></a>              <span class='hs-keyword'>let</span> <span class='hs-varid'>p'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varop'>$</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-varid'>ys</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-314"></a>              <span class='hs-keyword'>in</span> <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span> <span class='hs-varid'>x</span> <span class='hs-varid'>ys</span> <span class='hs-conid'>[]</span>
<a name="line-315"></a>              <span class='hs-varop'>++</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>us</span><span class='hs-layout'>)</span> <span class='hs-varid'>p'</span>
<a name="line-316"></a>          <span class='hs-varid'>graphAuts'</span> <span class='hs-varid'>us</span> <span class='hs-layout'>(</span><span class='hs-conid'>[]</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphAuts'</span> <span class='hs-varid'>us</span> <span class='hs-varid'>pt</span>
<a name="line-317"></a>          <span class='hs-varid'>graphAuts'</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-318"></a>          <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>ph</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-varid'>hs</span> <span class='hs-keyglyph'>=</span>
<a name="line-319"></a>              <span class='hs-keyword'>let</span> <span class='hs-varid'>px</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>x</span> <span class='hs-varid'>ph</span> <span class='hs-conop'>:</span> <span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-320"></a>                  <span class='hs-varid'>py</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ph</span> <span class='hs-conop'>:</span> <span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span>
<a name="line-321"></a>                  <span class='hs-varid'>uus</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zip</span> <span class='hs-varid'>us</span> <span class='hs-varid'>us</span>
<a name="line-322"></a>              <span class='hs-keyword'>in</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>dfsEquitable</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span><span class='hs-layout'>,</span><span class='hs-varid'>es'</span><span class='hs-layout'>,</span><span class='hs-varid'>nbrs_g</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>uus</span><span class='hs-layout'>)</span> <span class='hs-varid'>px</span> <span class='hs-varid'>py</span> <span class='hs-keyword'>of</span>
<a name="line-323"></a>                 <span class='hs-conid'>[]</span>  <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span> <span class='hs-varid'>x</span> <span class='hs-varid'>ys</span> <span class='hs-varid'>hs</span>
<a name="line-324"></a>                 <span class='hs-varid'>h</span><span class='hs-conop'>:</span><span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>hs'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>h</span><span class='hs-conop'>:</span><span class='hs-varid'>hs</span> <span class='hs-keyword'>in</span> <span class='hs-varid'>h</span> <span class='hs-conop'>:</span> <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>ys</span> <span class='hs-conid'>L</span><span class='hs-varop'>.\\</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-varop'>.^^</span> <span class='hs-varid'>hs'</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>hs'</span>
<a name="line-325"></a>          <span class='hs-varid'>level</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-326"></a>          <span class='hs-varid'>dps</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vs</span><span class='hs-keyglyph'>]</span>
<a name="line-327"></a>          <span class='hs-varid'>es'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varid'>es</span>
<a name="line-328"></a>          <span class='hs-varid'>nbrs_g</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>nbrs</span> <span class='hs-varid'>g</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vs</span><span class='hs-keyglyph'>]</span>
<a name="line-329"></a>
<a name="line-330"></a><a name="dfsEquitable"></a><span class='hs-definition'>dfsEquitable</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span><span class='hs-layout'>,</span><span class='hs-varid'>es'</span><span class='hs-layout'>,</span><span class='hs-varid'>nbrs_g</span><span class='hs-layout'>)</span> <span class='hs-varid'>xys</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>dfs</span> <span class='hs-varid'>xys</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span> <span class='hs-keyword'>where</span>
<a name="line-331"></a>    <span class='hs-varid'>dfs</span> <span class='hs-varid'>xys</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span>
<a name="line-332"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>p1</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>p2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-333"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span>
<a name="line-334"></a>            <span class='hs-keyword'>let</span> <span class='hs-varid'>p1'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varid'>p1</span>
<a name="line-335"></a>                <span class='hs-varid'>p2'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varid'>p2</span>
<a name="line-336"></a>                <span class='hs-layout'>(</span><span class='hs-varid'>p1e</span><span class='hs-layout'>,</span><span class='hs-varid'>p2e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toEquitable2</span> <span class='hs-varid'>nbrs_g</span> <span class='hs-varid'>p1'</span> <span class='hs-varid'>p2'</span>
<a name="line-337"></a>            <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>p1e</span>
<a name="line-338"></a>               <span class='hs-keyword'>then</span> <span class='hs-conid'>[]</span>
<a name="line-339"></a>               <span class='hs-keyword'>else</span>
<a name="line-340"></a>                   <span class='hs-keyword'>if</span> <span class='hs-varid'>all</span> <span class='hs-varid'>isSingleton</span> <span class='hs-varid'>p1e</span>
<a name="line-341"></a>                   <span class='hs-keyword'>then</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>xys'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>xys</span> <span class='hs-varop'>++</span> <span class='hs-varid'>zip</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>p1e</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>p2e</span><span class='hs-layout'>)</span>
<a name="line-342"></a>                        <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>isCompatible</span> <span class='hs-varid'>xys'</span> <span class='hs-keyword'>then</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>fromPairs'</span> <span class='hs-varid'>xys'</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>else</span> <span class='hs-conid'>[]</span>
<a name="line-343"></a>                   <span class='hs-keyword'>else</span> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>p1''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p1e</span>
<a name="line-344"></a>                            <span class='hs-varid'>ys</span><span class='hs-conop'>:</span><span class='hs-varid'>p2''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p2e</span>
<a name="line-345"></a>                        <span class='hs-keyword'>in</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>xys</span><span class='hs-layout'>)</span>
<a name="line-346"></a>                                       <span class='hs-layout'>(</span><span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-varid'>xs</span> <span class='hs-conop'>:</span> <span class='hs-varid'>p1''</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-347"></a>                                       <span class='hs-layout'>(</span><span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>p2''</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-348"></a>                                       <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-349"></a>    <span class='hs-varid'>isCompatible</span> <span class='hs-varid'>xys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>and</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>x'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span> <span class='hs-varop'>&lt;</span> <span class='hs-varid'>x'</span><span class='hs-keyglyph'>]</span>
<a name="line-350"></a>
<a name="line-351"></a>
<a name="line-352"></a><span class='hs-comment'>-- AUTS OF INCIDENCE STRUCTURE VIA INCIDENCE GRAPH</span>
<a name="line-353"></a>
<a name="line-354"></a><span class='hs-comment'>-- based on graphAuts as applied to the incidence graph, but modified to avoid point-block crossover auts</span>
<a name="line-355"></a>
<a name="line-356"></a><a name="incidenceAuts"></a><span class='hs-comment'>-- |Given the incidence graph of an incidence structure between points and blocks</span>
<a name="line-357"></a><span class='hs-comment'>-- (for example, a set system),</span>
<a name="line-358"></a><span class='hs-comment'>-- @incidenceAuts g@ returns generators for the automorphism group of the incidence structure.</span>
<a name="line-359"></a><span class='hs-comment'>-- The generators are represented as permutations of the points.</span>
<a name="line-360"></a><span class='hs-comment'>-- The incidence graph should be represented with the points on the left and the blocks on the right.</span>
<a name="line-361"></a><span class='hs-comment'>-- If the incidence graph is connected, then the generators will be a strong generating set.</span>
<a name="line-362"></a><span class='hs-definition'>incidenceAuts</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>p</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-layout'>(</span><span class='hs-conid'>Either</span> <span class='hs-varid'>p</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Permutation</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>]</span>
<a name="line-363"></a><span class='hs-definition'>incidenceAuts</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>autsWithinComponents</span> <span class='hs-varop'>++</span> <span class='hs-varid'>isosBetweenComponents</span>
<a name="line-364"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>cs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>inducedSubgraph</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>components</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span>
<a name="line-365"></a>          <span class='hs-varid'>autsWithinComponents</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>concatMap</span> <span class='hs-varid'>incidenceAutsCon</span> <span class='hs-varid'>cs</span>
<a name="line-366"></a>          <span class='hs-varid'>isosBetweenComponents</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-varid'>swapFromIso</span> <span class='hs-varop'>$</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>take</span> <span class='hs-num'>1</span> <span class='hs-layout'>(</span><span class='hs-varid'>incidenceIsos</span> <span class='hs-varid'>ci</span> <span class='hs-varid'>cj</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>ci</span><span class='hs-layout'>,</span><span class='hs-varid'>cj</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>pairs</span> <span class='hs-varid'>cs</span><span class='hs-keyglyph'>]</span>
<a name="line-367"></a>          <span class='hs-varid'>swapFromIso</span> <span class='hs-varid'>xys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromPairs</span> <span class='hs-layout'>(</span><span class='hs-varid'>xys</span> <span class='hs-varop'>++</span> <span class='hs-varid'>map</span> <span class='hs-varid'>swap</span> <span class='hs-varid'>xys</span><span class='hs-layout'>)</span>
<a name="line-368"></a>          <span class='hs-varid'>swap</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-369"></a>
<a name="line-370"></a><a name="incidenceAutsCon"></a><span class='hs-definition'>incidenceAutsCon</span> <span class='hs-varid'>g</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>G</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>es</span><span class='hs-layout'>)</span> 
<a name="line-371"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>isConnected</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-varid'>points</span> <span class='hs-layout'>(</span><span class='hs-varid'>incidenceAuts'</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>vs</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span>
<a name="line-372"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"incidenceAutsCon: graph is not connected"</span>
<a name="line-373"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>points</span> <span class='hs-varid'>h</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromPairs</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Left</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>toPairs</span> <span class='hs-varid'>h</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- filtering out the action on blocks</span>
<a name="line-374"></a>          <span class='hs-varid'>incidenceAuts'</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-375"></a>              <span class='hs-comment'>-- let p' = L.sort $ filter (not . null) $ refine' (ys:pt) (dps M.! x)</span>
<a name="line-376"></a>              <span class='hs-keyword'>let</span> <span class='hs-varid'>p'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varop'>$</span> <span class='hs-varid'>refine</span> <span class='hs-layout'>(</span><span class='hs-varid'>ys</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-377"></a>              <span class='hs-keyword'>in</span> <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span> <span class='hs-varid'>x</span> <span class='hs-varid'>ys</span> <span class='hs-conid'>[]</span>
<a name="line-378"></a>              <span class='hs-varop'>++</span> <span class='hs-varid'>incidenceAuts'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>us</span><span class='hs-layout'>)</span> <span class='hs-varid'>p'</span>
<a name="line-379"></a>          <span class='hs-varid'>incidenceAuts'</span> <span class='hs-varid'>us</span> <span class='hs-layout'>(</span><span class='hs-conid'>[]</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>incidenceAuts'</span> <span class='hs-varid'>us</span> <span class='hs-varid'>pt</span>
<a name="line-380"></a>          <span class='hs-varid'>incidenceAuts'</span> <span class='hs-keyword'>_</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-keyword'>_</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span> <span class='hs-comment'>-- if we fix all the points, then the blocks must be fixed too</span>
<a name="line-381"></a>          <span class='hs-varid'>incidenceAuts'</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-382"></a>          <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>ph</span><span class='hs-conop'>:</span><span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-varid'>hs</span> <span class='hs-keyglyph'>=</span>
<a name="line-383"></a>              <span class='hs-keyword'>let</span> <span class='hs-varid'>px</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>x</span> <span class='hs-varid'>ph</span> <span class='hs-conop'>:</span> <span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-384"></a>                  <span class='hs-varid'>py</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ph</span> <span class='hs-conop'>:</span> <span class='hs-varid'>pt</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span>
<a name="line-385"></a>                  <span class='hs-varid'>uus</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zip</span> <span class='hs-varid'>us</span> <span class='hs-varid'>us</span>
<a name="line-386"></a>              <span class='hs-keyword'>in</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>dfsEquitable</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps</span><span class='hs-layout'>,</span><span class='hs-varid'>es'</span><span class='hs-layout'>,</span><span class='hs-varid'>nbrs_g</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>uus</span><span class='hs-layout'>)</span> <span class='hs-varid'>px</span> <span class='hs-varid'>py</span> <span class='hs-keyword'>of</span>
<a name="line-387"></a>                 <span class='hs-conid'>[]</span>  <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span> <span class='hs-varid'>x</span> <span class='hs-varid'>ys</span> <span class='hs-varid'>hs</span>
<a name="line-388"></a>                 <span class='hs-varid'>h</span><span class='hs-conop'>:</span><span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>hs'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>h</span><span class='hs-conop'>:</span><span class='hs-varid'>hs</span> <span class='hs-keyword'>in</span> <span class='hs-varid'>h</span> <span class='hs-conop'>:</span> <span class='hs-varid'>level</span> <span class='hs-varid'>us</span> <span class='hs-varid'>p</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>ys</span> <span class='hs-conid'>L</span><span class='hs-varop'>.\\</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-varop'>.^^</span> <span class='hs-varid'>hs'</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>hs'</span>
<a name="line-389"></a>          <span class='hs-varid'>level</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span> <span class='hs-comment'>-- includes the case where y matches Right _, which can only occur on first level, before we've distance partitioned</span>
<a name="line-390"></a>          <span class='hs-varid'>dps</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vs</span><span class='hs-keyglyph'>]</span>
<a name="line-391"></a>          <span class='hs-varid'>es'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varid'>es</span>
<a name="line-392"></a>          <span class='hs-varid'>nbrs_g</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>nbrs</span> <span class='hs-varid'>g</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vs</span><span class='hs-keyglyph'>]</span>
<a name="line-393"></a>
<a name="line-394"></a>
<a name="line-395"></a><span class='hs-comment'>-- GRAPH ISOMORPHISMS</span>
<a name="line-396"></a>
<a name="line-397"></a><span class='hs-comment'>-- !! not yet using equitable partitions, so could probably be more efficient</span>
<a name="line-398"></a>
<a name="line-399"></a>
<a name="line-400"></a><a name="graphIsos"></a><span class='hs-comment'>-- graphIsos :: (Ord a, Ord b) =&gt; Graph a -&gt; Graph b -&gt; [[(a,b)]]</span>
<a name="line-401"></a><span class='hs-definition'>graphIsos</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>g2</span>
<a name="line-402"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>length</span> <span class='hs-varid'>cs1</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>cs2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-403"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>graphIsos'</span> <span class='hs-varid'>cs1</span> <span class='hs-varid'>cs2</span>
<a name="line-404"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>cs1</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>inducedSubgraph</span> <span class='hs-varid'>g1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>components</span> <span class='hs-varid'>g1</span><span class='hs-layout'>)</span>
<a name="line-405"></a>          <span class='hs-varid'>cs2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>inducedSubgraph</span> <span class='hs-varid'>g2</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>components</span> <span class='hs-varid'>g2</span><span class='hs-layout'>)</span>
<a name="line-406"></a>          <span class='hs-varid'>graphIsos'</span> <span class='hs-layout'>(</span><span class='hs-varid'>ci</span><span class='hs-conop'>:</span><span class='hs-varid'>cis</span><span class='hs-layout'>)</span> <span class='hs-varid'>cjs</span> <span class='hs-keyglyph'>=</span>
<a name="line-407"></a>              <span class='hs-keyglyph'>[</span><span class='hs-varid'>iso</span> <span class='hs-varop'>++</span> <span class='hs-varid'>iso'</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>cj</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>cjs</span><span class='hs-layout'>,</span>
<a name="line-408"></a>                             <span class='hs-varid'>iso</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>graphIsosCon</span> <span class='hs-varid'>ci</span> <span class='hs-varid'>cj</span><span class='hs-layout'>,</span>
<a name="line-409"></a>                             <span class='hs-keyword'>let</span> <span class='hs-varid'>cjs'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>cj</span> <span class='hs-varid'>cjs</span><span class='hs-layout'>,</span>
<a name="line-410"></a>                             <span class='hs-varid'>iso'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>graphIsos'</span> <span class='hs-varid'>cis</span> <span class='hs-varid'>cjs'</span><span class='hs-keyglyph'>]</span>
<a name="line-411"></a>          <span class='hs-varid'>graphIsos'</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>[]</span><span class='hs-keyglyph'>]</span>
<a name="line-412"></a>
<a name="line-413"></a><a name="graphIsosCon"></a><span class='hs-comment'>-- isos between connected graphs</span>
<a name="line-414"></a><span class='hs-definition'>graphIsosCon</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>g2</span> 
<a name="line-415"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>isConnected</span> <span class='hs-varid'>g1</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>isConnected</span> <span class='hs-varid'>g2</span>
<a name="line-416"></a>        <span class='hs-keyglyph'>=</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>dfs</span> <span class='hs-conid'>[]</span> <span class='hs-layout'>(</span><span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>v1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g2</span> <span class='hs-varid'>v2</span><span class='hs-layout'>)</span>
<a name="line-417"></a>                 <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v1</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>take</span> <span class='hs-num'>1</span> <span class='hs-layout'>(</span><span class='hs-varid'>vertices</span> <span class='hs-varid'>g1</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>v2</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vertices</span> <span class='hs-varid'>g2</span><span class='hs-keyglyph'>]</span>
<a name="line-418"></a>                 <span class='hs-comment'>-- the take 1 handles the case where g1 is the null graph</span>
<a name="line-419"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"graphIsosCon: either or both graphs are not connected"</span>
<a name="line-420"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>dfs</span> <span class='hs-varid'>xys</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span>
<a name="line-421"></a>              <span class='hs-keyglyph'>|</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>p1</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>p2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-422"></a>              <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span>
<a name="line-423"></a>                  <span class='hs-keyword'>let</span> <span class='hs-varid'>p1'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varid'>p1</span>
<a name="line-424"></a>                      <span class='hs-varid'>p2'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varid'>p2</span>
<a name="line-425"></a>                  <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>all</span> <span class='hs-varid'>isSingleton</span> <span class='hs-varid'>p1'</span>
<a name="line-426"></a>                     <span class='hs-keyword'>then</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>xys'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>xys</span> <span class='hs-varop'>++</span> <span class='hs-varid'>zip</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>p1'</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>p2'</span><span class='hs-layout'>)</span>
<a name="line-427"></a>                          <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>isCompatible</span> <span class='hs-varid'>xys'</span> <span class='hs-keyword'>then</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>xys'</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>else</span> <span class='hs-conid'>[]</span>
<a name="line-428"></a>                     <span class='hs-keyword'>else</span> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>p1''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p1'</span>
<a name="line-429"></a>                              <span class='hs-varid'>ys</span><span class='hs-conop'>:</span><span class='hs-varid'>p2''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p2'</span>
<a name="line-430"></a>                          <span class='hs-keyword'>in</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>xys</span><span class='hs-layout'>)</span>
<a name="line-431"></a>                                         <span class='hs-layout'>(</span><span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-varid'>xs</span> <span class='hs-conop'>:</span> <span class='hs-varid'>p1''</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps1</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-432"></a>                                         <span class='hs-layout'>(</span><span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>p2''</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps2</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-433"></a>                                         <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-434"></a>          <span class='hs-varid'>isCompatible</span> <span class='hs-varid'>xys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>and</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>x'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es1</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span> <span class='hs-varop'>&lt;</span> <span class='hs-varid'>x'</span><span class='hs-keyglyph'>]</span>
<a name="line-435"></a>          <span class='hs-varid'>dps1</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vertices</span> <span class='hs-varid'>g1</span><span class='hs-keyglyph'>]</span>
<a name="line-436"></a>          <span class='hs-varid'>dps2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g2</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vertices</span> <span class='hs-varid'>g2</span><span class='hs-keyglyph'>]</span>
<a name="line-437"></a>          <span class='hs-varid'>es1</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varop'>$</span> <span class='hs-varid'>edges</span> <span class='hs-varid'>g1</span>
<a name="line-438"></a>          <span class='hs-varid'>es2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varop'>$</span> <span class='hs-varid'>edges</span> <span class='hs-varid'>g2</span>
<a name="line-439"></a>
<a name="line-440"></a>
<a name="line-441"></a><a name="isGraphIso"></a><span class='hs-comment'>-- |Are the two graphs isomorphic?</span>
<a name="line-442"></a><span class='hs-definition'>isGraphIso</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-443"></a><span class='hs-definition'>isGraphIso</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>g2</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>graphIsos</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>g2</span><span class='hs-layout'>)</span>
<a name="line-444"></a><span class='hs-comment'>-- !! If we're only interested in seeing whether or not two graphs are iso,</span>
<a name="line-445"></a><span class='hs-comment'>-- !! then the cost of calculating distancePartitions may not be warranted</span>
<a name="line-446"></a><span class='hs-comment'>-- !! (see Math.Combinatorics.Poset: orderIsos01 versus orderIsos)</span>
<a name="line-447"></a>
<a name="line-448"></a><a name="isIso"></a><span class='hs-comment'>-- !! deprecate</span>
<a name="line-449"></a><span class='hs-definition'>isIso</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>g2</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>graphIsos</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>g2</span><span class='hs-layout'>)</span>
<a name="line-450"></a>
<a name="line-451"></a>
<a name="line-452"></a><span class='hs-comment'>-- the following differs from graphIsos in only two ways</span>
<a name="line-453"></a><span class='hs-comment'>-- we avoid Left, Right crossover isos, by insisting that a Left is taken to a Left (first two lines)</span>
<a name="line-454"></a><span class='hs-comment'>-- we return only the action on the Lefts, and unLeft it</span>
<a name="line-455"></a><span class='hs-comment'>-- incidenceIsos :: (Ord p1, Ord b1, Ord p2, Ord b2) =&gt;</span>
<a name="line-456"></a><span class='hs-comment'>--     Graph (Either p1 b1) -&gt; Graph (Either p2 b2) -&gt; [[(p1,p2)]]</span>
<a name="line-457"></a>
<a name="line-458"></a><a name="incidenceIsos"></a><span class='hs-definition'>incidenceIsos</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>g2</span>
<a name="line-459"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>length</span> <span class='hs-varid'>cs1</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>cs2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-460"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>incidenceIsos'</span> <span class='hs-varid'>cs1</span> <span class='hs-varid'>cs2</span>
<a name="line-461"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>cs1</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>inducedSubgraph</span> <span class='hs-varid'>g1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span> <span class='hs-varop'>.</span> <span class='hs-varid'>lefts</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>components</span> <span class='hs-varid'>g1</span><span class='hs-layout'>)</span>
<a name="line-462"></a>          <span class='hs-varid'>cs2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>inducedSubgraph</span> <span class='hs-varid'>g2</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span> <span class='hs-varop'>.</span> <span class='hs-varid'>lefts</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>components</span> <span class='hs-varid'>g2</span><span class='hs-layout'>)</span>
<a name="line-463"></a>          <span class='hs-varid'>incidenceIsos'</span> <span class='hs-layout'>(</span><span class='hs-varid'>ci</span><span class='hs-conop'>:</span><span class='hs-varid'>cis</span><span class='hs-layout'>)</span> <span class='hs-varid'>cjs</span> <span class='hs-keyglyph'>=</span>
<a name="line-464"></a>              <span class='hs-keyglyph'>[</span><span class='hs-varid'>iso</span> <span class='hs-varop'>++</span> <span class='hs-varid'>iso'</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>cj</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>cjs</span><span class='hs-layout'>,</span>
<a name="line-465"></a>                             <span class='hs-varid'>iso</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>incidenceIsosCon</span> <span class='hs-varid'>ci</span> <span class='hs-varid'>cj</span><span class='hs-layout'>,</span>
<a name="line-466"></a>                             <span class='hs-keyword'>let</span> <span class='hs-varid'>cjs'</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>cj</span> <span class='hs-varid'>cjs</span><span class='hs-layout'>,</span>
<a name="line-467"></a>                             <span class='hs-varid'>iso'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>incidenceIsos'</span> <span class='hs-varid'>cis</span> <span class='hs-varid'>cjs'</span><span class='hs-keyglyph'>]</span>
<a name="line-468"></a>          <span class='hs-varid'>incidenceIsos'</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>[]</span><span class='hs-keyglyph'>]</span>
<a name="line-469"></a>
<a name="line-470"></a><a name="incidenceIsosCon"></a><span class='hs-definition'>incidenceIsosCon</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>g2</span>
<a name="line-471"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>isConnected</span> <span class='hs-varid'>g1</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>isConnected</span> <span class='hs-varid'>g2</span>
<a name="line-472"></a>        <span class='hs-keyglyph'>=</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>dfs</span> <span class='hs-conid'>[]</span> <span class='hs-layout'>(</span><span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>v1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g2</span> <span class='hs-varid'>v2</span><span class='hs-layout'>)</span>
<a name="line-473"></a>                 <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v1</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>take</span> <span class='hs-num'>1</span> <span class='hs-layout'>(</span><span class='hs-varid'>vertices</span> <span class='hs-varid'>g1</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>v2</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vertices</span> <span class='hs-varid'>g2</span><span class='hs-keyglyph'>]</span>
<a name="line-474"></a>                 <span class='hs-comment'>-- g1 may have no vertices</span>
<a name="line-475"></a>    <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"incidenceIsos: one or both graphs not connected"</span>
<a name="line-476"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>dfs</span> <span class='hs-varid'>xys</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span>
<a name="line-477"></a>              <span class='hs-keyglyph'>|</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>p1</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>map</span> <span class='hs-varid'>length</span> <span class='hs-varid'>p2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-478"></a>              <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span>
<a name="line-479"></a>                  <span class='hs-keyword'>let</span> <span class='hs-varid'>p1'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varid'>p1</span>
<a name="line-480"></a>                      <span class='hs-varid'>p2'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-varid'>p2</span>
<a name="line-481"></a>                  <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>all</span> <span class='hs-varid'>isSingleton</span> <span class='hs-varid'>p1'</span>
<a name="line-482"></a>                     <span class='hs-keyword'>then</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>xys'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>xys</span> <span class='hs-varop'>++</span> <span class='hs-varid'>zip</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>p1'</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>p2'</span><span class='hs-layout'>)</span>
<a name="line-483"></a>                          <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>isCompatible</span> <span class='hs-varid'>xys'</span> <span class='hs-keyword'>then</span> <span class='hs-keyglyph'>[</span><span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Left</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys'</span><span class='hs-keyglyph'>]</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>else</span> <span class='hs-conid'>[]</span>
<a name="line-484"></a>                     <span class='hs-keyword'>else</span> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>p1''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p1'</span>
<a name="line-485"></a>                              <span class='hs-varid'>ys</span><span class='hs-conop'>:</span><span class='hs-varid'>p2''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p2'</span>
<a name="line-486"></a>                          <span class='hs-keyword'>in</span> <span class='hs-varid'>concat</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>dfs</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>xys</span><span class='hs-layout'>)</span>
<a name="line-487"></a>                                         <span class='hs-layout'>(</span><span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-varid'>xs</span> <span class='hs-conop'>:</span> <span class='hs-varid'>p1''</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps1</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-488"></a>                                         <span class='hs-layout'>(</span><span class='hs-varid'>refine'</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>delete</span> <span class='hs-varid'>y</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>p2''</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>dps2</span> <span class='hs-conid'>M</span><span class='hs-varop'>.!</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-489"></a>                                         <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-490"></a>          <span class='hs-varid'>isCompatible</span> <span class='hs-varid'>xys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>and</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>x'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es1</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>`</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>member</span><span class='hs-varop'>`</span> <span class='hs-varid'>es2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-layout'>,</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xys</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span> <span class='hs-varop'>&lt;</span> <span class='hs-varid'>x'</span><span class='hs-keyglyph'>]</span>
<a name="line-491"></a>          <span class='hs-varid'>dps1</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vertices</span> <span class='hs-varid'>g1</span><span class='hs-keyglyph'>]</span>
<a name="line-492"></a>          <span class='hs-varid'>dps2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>distancePartition</span> <span class='hs-varid'>g2</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>vertices</span> <span class='hs-varid'>g2</span><span class='hs-keyglyph'>]</span>
<a name="line-493"></a>          <span class='hs-varid'>es1</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varop'>$</span> <span class='hs-varid'>edges</span> <span class='hs-varid'>g1</span>
<a name="line-494"></a>          <span class='hs-varid'>es2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>fromList</span> <span class='hs-varop'>$</span> <span class='hs-varid'>edges</span> <span class='hs-varid'>g2</span>
<a name="line-495"></a>
<a name="line-496"></a><a name="isIncidenceIso"></a><span class='hs-comment'>-- |Are the two incidence structures represented by these incidence graphs isomorphic?</span>
<a name="line-497"></a><span class='hs-definition'>isIncidenceIso</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>p1</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>b1</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>p2</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>b2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span>
<a name="line-498"></a>     <span class='hs-conid'>Graph</span> <span class='hs-layout'>(</span><span class='hs-conid'>Either</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>b1</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Graph</span> <span class='hs-layout'>(</span><span class='hs-conid'>Either</span> <span class='hs-varid'>p2</span> <span class='hs-varid'>b2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-499"></a><span class='hs-definition'>isIncidenceIso</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>g2</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>null</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>incidenceIsos</span> <span class='hs-varid'>g1</span> <span class='hs-varid'>g2</span><span class='hs-layout'>)</span>
<a name="line-500"></a>
<a name="line-501"></a><span class='hs-comment'>{-
<a name="line-502"></a>removeGens x gs = removeGens' [] gs where
<a name="line-503"></a>    baseOrbit = x .^^ gs
<a name="line-504"></a>    removeGens' ls (r:rs) =
<a name="line-505"></a>        if x .^^ (ls++rs) == baseOrbit
<a name="line-506"></a>        then removeGens' ls rs
<a name="line-507"></a>        else removeGens' (r:ls) rs
<a name="line-508"></a>    removeGens' ls [] = reverse ls
<a name="line-509"></a>-- !! reverse is probably pointless
<a name="line-510"></a>
<a name="line-511"></a>
<a name="line-512"></a>-- !! DON'T THINK THIS IS WORKING PROPERLY
<a name="line-513"></a>-- eg graphAutsSGSNew $ toGraph ([1..7],[[1,3],[2,3],[3,4],[4,5],[4,6],[4,7]])
<a name="line-514"></a>-- returns [[[1,2]],[[5,6]],[[5,7,6]],[[6,7]]]
<a name="line-515"></a>-- whereas [[6,7]] was a Schreier generator, so shouldn't have been listed
<a name="line-516"></a>
<a name="line-517"></a>-- Using Schreier generators to seed the next level
<a name="line-518"></a>-- At the moment this is slower than the above
<a name="line-519"></a>-- (This could be modified to allow us to start the search with a known subgroup)
<a name="line-520"></a>graphAutsNew g@(G vs es) = graphAuts' [] [] [vs] where
<a name="line-521"></a>    graphAuts' us hs p@((x:ys):pt) =
<a name="line-522"></a>        let ys' = ys L.\\ (x .^^ hs) -- don't need to consider points which can already be reached from Schreier generators
<a name="line-523"></a>            hs' = level us p x ys' []
<a name="line-524"></a>            p' = L.sort $ filter (not . null) $ refine' (ys:pt) (dps M.! x)
<a name="line-525"></a>            reps = cosetRepsGx (hs'++hs) x
<a name="line-526"></a>            schreierGens = removeGens x $ schreierGeneratorsGx (x,reps) (hs'++hs)
<a name="line-527"></a>        in hs' ++ graphAuts' (x:us) schreierGens p'
<a name="line-528"></a>    graphAuts' us hs ([]:pt) = graphAuts' us hs pt
<a name="line-529"></a>    graphAuts' _ _ [] = []
<a name="line-530"></a>    level us p@(ph:pt) x (y:ys) hs =
<a name="line-531"></a>        let px = refine' (L.delete x ph : pt) (dps M.! x)
<a name="line-532"></a>            py = refine' (L.delete y ph : pt) (dps M.! y)
<a name="line-533"></a>            uus = zip us us
<a name="line-534"></a>        in if map length px /= map length py
<a name="line-535"></a>           then level us p x ys hs
<a name="line-536"></a>           else case dfs ((x,y):uus) (filter (not . null) px) (filter (not . null) py) of
<a name="line-537"></a>                []  -&gt; level us p x ys hs
<a name="line-538"></a>                h:_ -&gt; let hs' = h:hs in h : level us p x (ys L.\\ (x .^^ hs')) hs'
<a name="line-539"></a>                -- if h1 = (1 2)(3 4), and h2 = (1 3 2), then we can remove 4 too
<a name="line-540"></a>    level _ _ _ [] _ = []
<a name="line-541"></a>    dfs xys p1 p2
<a name="line-542"></a>        | map length p1 /= map length p2 = []
<a name="line-543"></a>        | otherwise =
<a name="line-544"></a>            let p1' = filter (not . null) p1
<a name="line-545"></a>                p2' = filter (not . null) p2
<a name="line-546"></a>            in if all isSingleton p1'
<a name="line-547"></a>               then let xys' = xys ++ zip (concat p1') (concat p2')
<a name="line-548"></a>                    in if isCompatible xys' then [fromPairs' xys'] else []
<a name="line-549"></a>               else let (x:xs):p1'' = p1'
<a name="line-550"></a>                        ys:p2'' = p2'
<a name="line-551"></a>                    in concat [dfs ((x,y):xys)
<a name="line-552"></a>                                   (refine' (xs : p1'') (dps M.! x))
<a name="line-553"></a>                                   (refine' ((L.delete y ys):p2'') (dps M.! y))
<a name="line-554"></a>                                   | y &lt;- ys]
<a name="line-555"></a>    isCompatible xys = and [([x,x'] `S.member` es') == (L.sort [y,y'] `S.member` es') | (x,y) &lt;- xys, (x',y') &lt;- xys, x &lt; x']
<a name="line-556"></a>    dps = M.fromList [(v, distancePartition g v) | v &lt;- vs]
<a name="line-557"></a>    es' = S.fromList es
<a name="line-558"></a>-}</span>
<a name="line-559"></a>
<a name="line-560"></a>
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