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<h1>Digraphs</h1>
<h2>Graphs, digraphs, and multidigraphs in <strong class="pkg">GAP</strong></h2>
<p>
1.13.1</p>
<p>
27 September 2025
</p>
</div>
<p><b>
Jan De Beule
</b>
<br />Email: <span class="URL"><a href="mailto:jdebeule@cage.ugent.be">jdebeule@cage.ugent.be</a></span>
<br />Homepage: <span class="URL"><a href="https://researchportal.vub.be/en/persons/jan-de-beule">https://researchportal.vub.be/en/persons/jan-de-beule</a></span>
<br />Address: <br />Vrije Universiteit Brussel, Vakgroep Wiskunde, Pleinlaan 2, B - 1050 Brussels, Belgium<br />
</p><p><b>
Julius Jonusas
</b>
<br />Email: <span class="URL"><a href="mailto:j.jonusas@gmail.com">j.jonusas@gmail.com</a></span>
<br />Homepage: <span class="URL"><a href="http://julius.jonusas.work">http://julius.jonusas.work</a></span>
</p><p><b>
James Mitchell
</b>
<br />Email: <span class="URL"><a href="mailto:jdm3@st-andrews.ac.uk">jdm3@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://jdbm.me">https://jdbm.me</a></span>
<br />Address: <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />
</p><p><b>
Wilf A. Wilson
</b>
<br />Email: <span class="URL"><a href="mailto:gap@wilf-wilson.net">gap@wilf-wilson.net</a></span>
<br />Homepage: <span class="URL"><a href="https://wilf.me">https://wilf.me</a></span>
</p><p><b>
Michael Young
</b>
<br />Email: <span class="URL"><a href="mailto:mct25@st-andrews.ac.uk">mct25@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://myoung.uk/work/">https://myoung.uk/work/</a></span>
<br />Address: <br />Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<br />
</p><p><b>
Marina Anagnostopoulou-Merkouri
</b>
<br />Email: <span class="URL"><a href="mailto:mam49@st-andrews.ac.uk">mam49@st-andrews.ac.uk</a></span>
<br />Address: <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />
</p><p><b>
Finn Buck
</b>
<br />Email: <span class="URL"><a href="mailto:finneganlbuck@gmail.com">finneganlbuck@gmail.com</a></span>
<br />Address: <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />
</p><p><b>
Stuart Burrell
</b>
<br />Email: <span class="URL"><a href="mailto:stuartburrell1994@gmail.com">stuartburrell1994@gmail.com</a></span>
<br />Homepage: <span class="URL"><a href="https://stuartburrell.github.io">https://stuartburrell.github.io</a></span>
</p><p><b>
Graham Campbell
</b>
</p><p><b>
Raiyan Chowdhury
</b>
</p><p><b>
Reinis Cirpons
</b>
<br />Email: <span class="URL"><a href="mailto:rc234@st-andrews.ac.uk">rc234@st-andrews.ac.uk</a></span>
<br />Address: <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />
</p><p><b>
Ashley Clayton
</b>
<br />Email: <span class="URL"><a href="mailto:ac323@st-andrews.ac.uk">ac323@st-andrews.ac.uk</a></span>
<br />Address: <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />
</p><p><b>
Tom Conti-Leslie
</b>
<br />Email: <span class="URL"><a href="mailto:tom.contileslie@gmail.com">tom.contileslie@gmail.com</a></span>
<br />Homepage: <span class="URL"><a href="https://tomcontileslie.com">https://tomcontileslie.com</a></span>
</p><p><b>
Joseph Edwards
</b>
<br />Email: <span class="URL"><a href="mailto:jde1@st-andrews.ac.uk">jde1@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://github.com/Joseph-Edwards">https://github.com/Joseph-Edwards</a></span>
<br />Address: <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />
</p><p><b>
Luna Elliott
</b>
<br />Email: <span class="URL"><a href="mailto:luna.elliott142857@gmail.com">luna.elliott142857@gmail.com</a></span>
<br />Homepage: <span class="URL"><a href="https://research.manchester.ac.uk/en/persons/luna-elliott">https://research.manchester.ac.uk/en/persons/luna-elliott</a></span>
</p><p><b>
Jan Engelhardt
</b>
<br />Email: <span class="URL"><a href="mailto:jengelh@inai.de">jengelh@inai.de</a></span>
<br />Homepage: <span class="URL"><a href="https://inai.de">https://inai.de</a></span>
</p><p><b>
Isuru Fernando
</b>
<br />Email: <span class="URL"><a href="mailto:isuruf@gmail.com">isuruf@gmail.com</a></span>
</p><p><b>
Ewan Gilligan
</b>
<br />Email: <span class="URL"><a href="mailto:eg207@st-andrews.ac.uk">eg207@st-andrews.ac.uk</a></span>
</p><p><b>
Gillis Frankie
</b>
<br />Email: <span class="URL"><a href="mailto:fotg1@st-andrews.ac.uk">fotg1@st-andrews.ac.uk</a></span>
</p><p><b>
Sebastian Gutsche
</b>
<br />Email: <span class="URL"><a href="mailto:gutsche@momo.math.rwth-aachen.de">gutsche@momo.math.rwth-aachen.de</a></span>
</p><p><b>
Samantha Harper
</b>
<br />Email: <span class="URL"><a href="mailto:seh25@st-andrews.ac.uk">seh25@st-andrews.ac.uk</a></span>
</p><p><b>
Max Horn
</b>
<br />Email: <span class="URL"><a href="mailto:mhorn@rptu.de">mhorn@rptu.de</a></span>
<br />Homepage: <span class="URL"><a href="https://www.quendi.de/math">https://www.quendi.de/math</a></span>
<br />Address: <br />Fachbereich Mathematik, RPTU Kaiserslautern-Landau, Gottlieb-Daimler-Straße 48, 67663 Kaiserslautern, Germany<br />
</p><p><b>
Harry Jack
</b>
<br />Email: <span class="URL"><a href="mailto:hrj4@st-andrews.ac.uk">hrj4@st-andrews.ac.uk</a></span>
</p><p><b>
Christopher Jefferson
</b>
<br />Email: <span class="URL"><a href="mailto:caj21@st-andrews.ac.uk">caj21@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://heather.cafe/">https://heather.cafe/</a></span>
<br />Address: <br />Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<br />
</p><p><b>
Malachi Johns
</b>
<br />Email: <span class="URL"><a href="mailto:zlj1@st-andrews.ac.uk">zlj1@st-andrews.ac.uk</a></span>
</p><p><b>
Olexandr Konovalov
</b>
<br />Email: <span class="URL"><a href="mailto:obk1@st-andrews.ac.uk">obk1@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://olexandr-konovalov.github.io/">https://olexandr-konovalov.github.io/</a></span>
<br />Address: <br />Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<br />
</p><p><b>
Hyeokjun Kwon
</b>
<br />Email: <span class="URL"><a href="mailto:hk78@st-andrews.ac.uk">hk78@st-andrews.ac.uk</a></span>
</p><p><b>
Aidan Lau
</b>
</p><p><b>
Andrea Lee
</b>
<br />Email: <span class="URL"><a href="mailto:ahwl1@st-andrews.ac.uk">ahwl1@st-andrews.ac.uk</a></span>
</p><p><b>
Saffron McIver
</b>
<br />Email: <span class="URL"><a href="mailto:sm544@st-andrews.ac.uk">sm544@st-andrews.ac.uk</a></span>
</p><p><b>
Seyyed Ali Mohammadiyeh
</b>
<br />Email: <span class="URL"><a href="mailto:MaxBaseCode@Gmail.Com">MaxBaseCode@Gmail.Com</a></span>
</p><p><b>
Michael Orlitzky
</b>
<br />Email: <span class="URL"><a href="mailto:michael@orlitzky.com">michael@orlitzky.com</a></span>
<br />Homepage: <span class="URL"><a href="https://michael.orlitzky.com/">https://michael.orlitzky.com/</a></span>
</p><p><b>
Matthew Pancer
</b>
<br />Email: <span class="URL"><a href="mailto:mp322@st-andrews.ac.uk">mp322@st-andrews.ac.uk</a></span>
</p><p><b>
Markus Pfeiffer
</b>
<br />Email: <span class="URL"><a href="mailto:markus.pfeiffer@morphism.de">markus.pfeiffer@morphism.de</a></span>
<br />Homepage: <span class="URL"><a href="https://markusp.morphism.de/">https://markusp.morphism.de/</a></span>
</p><p><b>
Daniel Pointon
</b>
<br />Email: <span class="URL"><a href="mailto:dp211@st-andrews.ac.uk">dp211@st-andrews.ac.uk</a></span>
</p><p><b>
Pramoth Ragavan
</b>
<br />Email: <span class="URL"><a href="mailto:107881923+pramothragavan@users.noreply.github.com">107881923+pramothragavan@users.noreply.github.com</a></span>
</p><p><b>
Lea Racine
</b>
<br />Email: <span class="URL"><a href="mailto:lr217@st-andrews.ac.uk">lr217@st-andrews.ac.uk</a></span>
<br />Address: <br />Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<br />
</p><p><b>
Christopher Russell
</b>
</p><p><b>
Artur Schaefer
</b>
<br />Email: <span class="URL"><a href="mailto:as305@st-and.ac.uk">as305@st-and.ac.uk</a></span>
</p><p><b>
Isabella Scott
</b>
<br />Email: <span class="URL"><a href="mailto:iscott@uchicago.edu">iscott@uchicago.edu</a></span>
</p><p><b>
Kamran Sharma
</b>
<br />Email: <span class="URL"><a href="mailto:kks4@st-andrews.ac.uk">kks4@st-andrews.ac.uk</a></span>
<br />Address: <br />Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<br />
</p><p><b>
Finn Smith
</b>
<br />Email: <span class="URL"><a href="mailto:fls3@st-andrews.ac.uk">fls3@st-andrews.ac.uk</a></span>
<br />Address: <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />
</p><p><b>
Ben Spiers
</b>
<br />Email: <span class="URL"><a href="mailto:bspiers972@outlook.com">bspiers972@outlook.com</a></span>
</p><p><b>
Maria Tsalakou
</b>
<br />Email: <span class="URL"><a href="mailto:mt200@st-andrews.ac.uk">mt200@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://mariatsalakou.github.io/">https://mariatsalakou.github.io/</a></span>
<br />Address: <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />
</p><p><b>
Agastyaa Vishvanath
</b>
<br />Email: <span class="URL"><a href="mailto:av215@st-andrews.ac.uk">av215@st-andrews.ac.uk</a></span>
</p><p><b>
Meike Weiss
</b>
<br />Email: <span class="URL"><a href="mailto:weiss@art.rwth-aachen.de">weiss@art.rwth-aachen.de</a></span>
<br />Homepage: <span class="URL"><a href="https://bit.ly/4e6pUeP">https://bit.ly/4e6pUeP</a></span>
<br />Address: <br />Chair of Algebra and Representation Theory, Pontdriesch 10-16, 52062 Aachen<br />
</p><p><b>
Murray Whyte
</b>
<br />Email: <span class="URL"><a href="mailto:mw231@st-andrews.ac.uk">mw231@st-andrews.ac.uk</a></span>
<br />Address: <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />
</p><p><b>
Fabian Zickgraf
</b>
<br />Email: <span class="URL"><a href="mailto:f.zickgraf@dashdos.com">f.zickgraf@dashdos.com</a></span>
</p>
<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>The <strong class="pkg">Digraphs</strong> package is a <strong class="pkg">GAP</strong> package containing methods for graphs, digraphs, and multidigraphs.</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>Jan De Beule, Julius Jonušas, James D. Mitchell, Wilf A. Wilson, Michael Young et al.</p>
<p><strong class="pkg">Digraphs</strong> is free software; you can redistribute it and/or modify it under the terms of the <span class="URL"><a href=" https://www.fsf.org/licenses/gpl.html">GNU General Public License</a></span> as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>We would like to thank Christopher Jefferson for his help in including <span class="URL"><a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a></span> in <strong class="pkg">Digraphs</strong>. We also gratefully acknowledge the encouragement and assistance of Leonard Soicher, and the inspiration of his <span class="URL"><a href="https://gap-packages.github.io/grape">GRAPE</a></span> package, at many points throughout the development of <strong class="pkg">Digraphs</strong>. This package's methods for computing digraph homomorphisms are based on work by Max Neunhöffer, and independently Artur Schäfer.
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>
<div class="ContChap"><a href="chap1.html#X7F202ABA780E595D">1 <span class="Heading">
The <strong class="pkg">Digraphs</strong> package
</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7DFB63A97E67C0A1">1.1 <span class="Heading">Introduction</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X84541F61810C741D">1.1-1 <span class="Heading">Definitions</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap2.html#X817088B27F90D596">2 <span class="Heading">Installing <strong class="pkg">Digraphs</strong></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7DA3059C79842BF3">2.1 <span class="Heading">For those in a hurry</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7C1F2E6D860DBDEC">2.1-1 <span class="Heading">Configuration options</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X780AA9D97EBCA95D">2.2 <span class="Heading">Optional package dependencies</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8493C7587FCF6D8B">2.2-1 <span class="Heading">The Grape package</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X849F6196875A6DF5">2.3 <span class="Heading">Compiling the kernel module</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X857CBE5484CF703A">2.4 <span class="Heading">Rebuilding the documentation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X870631C38610AC25">2.4-1 DigraphsMakeDoc</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7862D3F37C5BBDEF">2.5 <span class="Heading">Testing your installation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X86AF4DAE80B978DA">2.5-1 DigraphsTestInstall</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A20088B7C406C4A">2.5-2 DigraphsTestStandard</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7885EF3E785D4298">2.5-3 DigraphsTestExtreme</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap3.html#X7D34861E863A5D93">3 <span class="Heading">Creating digraphs</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7D34861E863A5D93">3.1 <span class="Heading">Creating digraphs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7877ADC77F85E630">3.1-1 IsDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7D7EDF83820ED6F5">3.1-2 IsMutableDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7CAFAA89804F80BD">3.1-3 IsImmutableDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E749324800B38A5">3.1-4 IsCayleyDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X80F1B6D28478D8B9">3.1-5 IsDigraphWithAdjacencyFunction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86E798B779515678">3.1-6 DigraphByOutNeighboursType</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X834843057CE86655">3.1-7 Digraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8023FE387A3AB609">3.1-8 DigraphByAdjacencyMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7F37B6768349E269">3.1-9 DigraphByEdges</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B75C1D680757D6F">3.1-10 EdgeOrbitsDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X81BC49B57EAADEFB">3.1-11 DigraphByInNeighbours</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7FCADADC7EC28478">3.1-12 CayleyDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7BB820C9813F035F">3.1-13 ListNamedDigraphs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X83608C407CC8836D">3.2 <span class="Heading">Changing representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7FBC4BDB82FBEDD2">3.2-1 AsBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86834E307EACC670">3.2-2 AsDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B335342839E5146">3.2-3 Graph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C4F13E080EC16B0">3.2-4 AsGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C5360B2799943F3">3.2-5 AsTransformation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X85126078848B420A">3.3 <span class="Heading">New digraphs from old</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X83D93A8A8251E6F9">3.3-1 DigraphImmutableCopy</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8399F7427B227228">3.3-2 DigraphImmutableCopyIfImmutable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X83C51DA182CCEA2F">3.3-3 InducedSubdigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7CAF093B85A93D2F">3.3-4 ReducedDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X829E3EAC7C4B3B1E">3.3-5 MaximalSymmetricSubdigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79BA6A66846D5A95">3.3-6 MaximalAntiSymmetricSubdigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DD9766C86D3ED20">3.3-7 UndirectedSpanningForest</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79C770918610AD97">3.3-8 DigraphShortestPathSpanningTree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7D1D26D27F5B56C2">3.3-9 QuotientDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78DECD26811EFD7C">3.3-10 DigraphReverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7F71D99D852B130F">3.3-11 DigraphDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X874883DD7DD450C4">3.3-12 DigraphSymmetricClosure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A6C419080AD41DE">3.3-13 DigraphTransitiveClosure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82AD17517E273600">3.3-14 DigraphTransitiveReduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X83B2506D79453208">3.3-15 DigraphAddVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8134BEE7786BD3A7">3.3-16 DigraphAddVertices</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E58CC4880627658">3.3-17 DigraphAddEdge</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7BE5C7028760B053">3.3-18 DigraphAddEdgeOrbit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8693A61B7F752C76">3.3-19 DigraphAddEdges</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B634A2B83C08B16">3.3-20 DigraphRemoveVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E290E847A5A299A">3.3-21 DigraphRemoveVertices</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8433E3BC7E5EA6BF">3.3-22 DigraphRemoveEdge</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X85981D9187F49018">3.3-23 DigraphRemoveEdgeOrbit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X87093FDA7F88E732">3.3-24 DigraphRemoveEdges</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79324AF7818C0C02">3.3-25 DigraphRemoveLoops</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DCCD0247897A3DE">3.3-26 DigraphRemoveAllMultipleEdges</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X792AD1147E2BFCB7">3.3-27 DigraphContractEdge</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7821753D85402A8C">3.3-28 DigraphReverseEdges</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X814F1DFC83DB273F">3.3-29 DigraphDisjointUnion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DA997697D310E44">3.3-30 DigraphEdgeUnion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DDFC759860E3390">3.3-31 DigraphJoin</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7D625CC87DBFFDED">3.3-32 DigraphCartesianProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84D24DC9833B54A5">3.3-33 DigraphDirectProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8160BCC378AF000F">3.3-34 ConormalProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79FD2AF279F20A72">3.3-35 HomomorphicProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8151176882BA9901">3.3-36 LexicographicProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X807F95057F9DF576">3.3-37 ModularProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82E200F07FEFAF27">3.3-38 StrongProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8795087A78FE7D54">3.3-39 DigraphCartesianProductProjections</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84FB64F185B804C2">3.3-40 DigraphDirectProductProjections</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8595BF937B749F22">3.3-41 LineDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8364C6F17A1680CB">3.3-42 LineUndirectedDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7FB8B48C87C0ED16">3.3-43 DoubleDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C6E6CB284982C7A">3.3-44 BipartiteDoubleDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8167A50A83256ED1">3.3-45 DigraphAddAllLoops</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X865436437DF95FEF">3.3-46 DistanceDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86F9CCEA839ABC48">3.3-47 DigraphClosure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B5AC5FE859F4D80">3.3-48 DigraphMycielskian</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X85B5D9B97F5187B7">3.4 <span class="Heading">Random digraphs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86CF9F66788B2A24">3.4-1 RandomDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78FE275E7E77D56F">3.4-2 RandomMultiDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7D36B5E57F055051">3.4-3 RandomTournament</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A023E4787682475">3.4-4 RandomLattice</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7C76D1DC7DAF03D3">3.5 <span class="Heading">Standard examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7D5E3E337D03EDFF">3.5-1 AndrasfaiGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8636C2898395B7DF">3.5-2 BananaTree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7FC309427BB170D8">3.5-3 BinaryTree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X845C374280D6EAA4">3.5-4 BinomialTreeGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E3240047C92733F">3.5-5 BishopsGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X81A0AC37816D287B">3.5-6 BondyGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X861F493382FA7C0B">3.5-7 BookGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8542287E81BDB55E">3.5-8 BurntPancakeGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84C1D3D67E3979A5">3.5-9 PancakeGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B5E9D857D47F5C2">3.5-10 StackedBookGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X870594FC866AC88E">3.5-11 ChainDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DB5AB657A797CF2">3.5-12 CirculantGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X812417E278198D9C">3.5-13 CompleteDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8795B0AD856014FA">3.5-14 CompleteBipartiteDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X873F29CC863241F8">3.5-15 CompleteMultipartiteDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X80C29DDE876FFBEB">3.5-16 CycleDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7F6EC0AE81531C3C">3.5-17 CycleGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X80DAE31A79FEFD40">3.5-18 EmptyDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7CE45E2B782ADE9A">3.5-19 GearGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X795B62398767E313">3.5-20 HaarGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X801024F57DDC8A39">3.5-21 HalvedCubeGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C43BDE47DF6553A">3.5-22 HanoiGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X782ABFCE812B020A">3.5-23 HelmGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7EE552F88609B1A2">3.5-24 HypercubeGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X80ED9CE785819607">3.5-25 JohnsonDigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79483C677AF65688">3.5-26 KellerGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X80576A8C861512FD">3.5-27 KingsGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8655BA8584B3ACD0">3.5-28 KneserGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84609DCA79FD9B56">3.5-29 KnightsGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7F61140C822880DA">3.5-30 LindgrenSousselierGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X832E82CF87BF5D43">3.5-31 LollipopGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X853613228110588E">3.5-32 MobiusLadderGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X825943547FD7A687">3.5-33 MycielskiGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7904BB2982014ADA">3.5-34 OddGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X815055168405B7F0">3.5-35 PathGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A38DFC47AAE4A96">3.5-36 PermutationStarGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X823F43217A6C375D">3.5-37 PetersenGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B5441F386BD105E">3.5-38 GeneralisedPetersenGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X85425DC5847E6D20">3.5-39 PrismGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X817982877B48D5BD">3.5-40 StackedPrismGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X785C3F1F7D690151">3.5-41 QueensGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A6DD11881874F51">3.5-42 RooksGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8404987F849D7CF2">3.5-43 SquareGridGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8234361278E8816F">3.5-44 TriangularGridGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78F78C077CBAE1EC">3.5-45 StarGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X81D59C4D809F4AD3">3.5-46 TadpoleGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7BFA33067F83F8B0">3.5-47 WalshHadamardGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84F3B70A82EEE780">3.5-48 WebGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X817EA60D828A765E">3.5-49 WheelGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7BE44CA27AA5F8DB">3.5-50 WindmillGraph</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4.html#X7AD6F77E7D95C996">4 <span class="Heading">Operators</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X84E0B5B88358C96B">4.1 <span class="Heading">Operators for digraphs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X829B911D7EFD2D85">4.1-1 IsSubdigraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X833C3299787E2309">4.1-2 IsUndirectedSpanningTree</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X8739F6CD78C90B14">5 <span class="Heading">Attributes and operations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7E814B6478F7D015">5.1 <span class="Heading">Vertices and edges</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7C45F7D878D896AC">5.1-1 DigraphVertices</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7C6F19B57CB2E882">5.1-2 DigraphNrVertices</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7D1C6A4D7ECEC317">5.1-3 DigraphEdges</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X85E1CFDD7E164AD0">5.1-4 DigraphNrEdges</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7BD5D255809C859E">5.1-5 DigraphNrAdjacencies</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7E53E728795BB862">5.1-6 DigraphNrAdjacenciesWithoutLoops</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7BDABAF07917462B">5.1-7 DigraphNrLoops</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X85D5E08280914EE4">5.1-8 DigraphSinks</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F5C6268839BE98C">5.1-9 DigraphSources</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X785C30378064CF47">5.1-10 DigraphTopologicalSort</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7CA91E4B7904F793">5.1-11 DigraphVertexLabel</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7E51F2FE87140B32">5.1-12 DigraphVertexLabels</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X79FAEACC7F438C2F">5.1-13 DigraphEdgeLabel</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7C24851087D4A8FB">5.1-14 DigraphEdgeLabels</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7EFAF01B7A155157">5.1-15 DigraphInEdges</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7BECFE6687ECD028">5.1-16 DigraphOutEdges</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7BB8ED88835F07B4">5.1-17 IsDigraphEdge</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X848FED0B7B4ACD1F">5.1-18 IsMatching</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7B7A67277B1C9A02">5.1-19 DigraphMaximalMatching</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X78E9847A858788D1">5.1-20 DigraphMaximumMatching</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7D7CE8328187D0DF">5.2 <span class="Heading">Neighbours and degree</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7DC2CD70830BEE60">5.2-1 AdjacencyMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X87FA0A727CDB060B">5.2-2 CharacteristicPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8507DC4F794780C1">5.2-3 BooleanAdjacencyMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7AFCE34A7A04D5C1">5.2-4 DigraphAdjacencyFunction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7FDEBF3279759961">5.2-5 DigraphRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7E9880767AE68E00">5.2-6 OutNeighbours</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X85C7AA5A81DA6E11">5.2-7 InNeighbours</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F5ACE807D1BC2E2">5.2-8 OutDegrees</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7ADDFBFD7A365775">5.2-9 InDegrees</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7A09EB648070276D">5.2-10 OutDegreeOfVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X83315B0186850806">5.2-11 OutNeighboursOfVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7C9CD0527CB9E6EF">5.2-12 InDegreeOfVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7C0DA18B8291F302">5.2-13 InNeighboursOfVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X83271F607BD809CF">5.2-14 DigraphLoops</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7BEAE1C78267F54D">5.2-15 DegreeMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X865390B08331936B">5.2-16 LaplacianMatrix</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X86424F167BD4F629">5.3 <span class="Heading">Orders</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7DDB33B686B3A2C6">5.3-1 PartialOrderDigraphMeetOfVertices</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X824B9896798530F6">5.3-2 NonUpperSemimodularPair</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8537F4088400DC48">5.4 <span class="Heading">Reachability and connectivity</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F16B9EB8398459C">5.4-1 DigraphDiameter</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8104A9D37BCD8A05">5.4-2 DigraphShortestDistance</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X81F99BC67E9D050F">5.4-3 DigraphShortestDistances</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8223718079D98A82">5.4-4 DigraphLongestDistanceFromVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7CB7DDCD84621D38">5.4-5 DigraphDistanceSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X79A3DA4078CF3C90">5.4-6 DigraphGirth</a></ | |
