Graphs and paths
$\gdef\badDispatch#1{\textbf{\textcolor{e1432d}{#1}}} \gdef\noKatexForm#1{\badDispatch{#1}} \gdef\largeDot{\raisebox{0.06em}{\tiny∙}} \gdef\rarrbar{{\raisebox{-0.05em}{→}\mkern{-0.13em}{\large\shortmid}}} \gdef\larrbar{{{\large\shortmid}\mkern{-0.13em}{\raisebox{-0.05em}{←}}}} \gdef\suptrans{^\mathsf{T}} \gdef\supdagger{^\dagger} \gdef\cardinalRewrite#1#2{\rewrite{#1}{#2}} \gdef\primed#1{{#1}^{\prime}} \gdef\tinybullet{{\tiny●}} \gdef\factsym#1{\mathord{#1}} \gdef\forwardFactor{\factsym{\uparrow}} \gdef\backwardFactor{\factsym{\downarrow}} \gdef\forwardBackwardFactor{\factsym{\updownarrow}} \gdef\forwardBackwardNeutralFactor{\factsym{\mathrlap{\downarrow}{\mathrlap{\uparrow}{\endash} } }} \gdef\neutralFactor{\factsym{\shornarrow}} \gdef\edgedot{↫} \gdef\desym{\mathbin{↣}} \gdef\uesym{\mathbin{↢}} \gdef\de#1#2{{{#1}\desym{#2}}} \gdef\ue#1#2{{{#1}\uesym{#2}}} \gdef\shifttag#1{\raisebox{-1em}{#1}} \gdef\shifttag#1{#1} \gdef\tde#1#2#3{{#1}\:\xrightedge{\shifttag{#3}}\;{#2}} \gdef\tue#1#2#3{{#1}\;\xundirectededge{\shifttag{#3}}\;{#2}} \gdef\shiftunderset#1#2{\underset{\raisebox{0.15em}{\scriptsize #1}}{#2}} \gdef\mdtde#1#2#3#4{{#1}\,\;\shiftunderset{#4}{\xrightedge{#3}}\;{#2}} \gdef\mtue#1#2#3#4{{#1}\,\;\shiftunderset{#4}{\xundirectededge{#3}}\;\,{#2}} \gdef\mtde#1#2#3#4{{#1}\,\;\operatornamewithlimits{\xrightedge{#3}}\limits_{#4} \;{#2}} \gdef\mapsfrom{\htmlClass{hreflect}{\mapsto}} \gdef\longmapsfrom{\htmlClass{hreflect}{\longmapsto}} \gdef\diffd{𝕕} \gdef\partialdof#1{\partial {#1}} \gdef\textAnd{\text{\,and\,}} \gdef\identicallyEqualSymbol{\equiv} \gdef\congruentSymbol{\equiv} \gdef\isomorphicSymbol{\simeq} \gdef\homeomorphicSymbol{\cong} \gdef\homotopicSymbol{\simeq} \gdef\approxEqualSymbol{\approx} \gdef\bijectiveSymbol{\approx} \gdef\defeq{\mathrel{≝}} \gdef\defEqualSymbol{\mathrel{≝}} \gdef\syntaxEqualSymbol{\mathrel{\textcolor{888888}{\equiv}}} \gdef\tailEqualSymbol{\underdot{=}} \gdef\headEqualSymbol{\dot{=}} \gdef\ruledelayed{:\to} \gdef\mathdollar{\text{\textdollar}} \gdef\hyphen{\text{-}} \gdef\endash{\text{--}} \gdef\emdash{\text{---}} \gdef\updownarrows{\uparrow\!\downarrow} \gdef\vthinspace{\mkern{1pt}} \gdef\dlq{\text{\textquotedblleft}} \gdef\drq{\text{\textquotedblright}} \gdef\dprime{{\prime\prime}} \gdef\inverse#1{{{#1}^{-1}}} \gdef\inverseSymbol{\inverse{□}} \gdef\groupDirectProduct#1{#1} \gdef\groupDirectProductSymbol{\times} \gdef\groupInverse{\inverse} \gdef\groupPower#1#2{{#1}^{#2}} \gdef\groupCommutator#1#2{\left[{#1},{#2}\right]} \gdef\groupoidInverse{\inverse} \gdef\latticeBFS#1{\textrm{bfs}({#1})} \gdef\pathHomomorphism#1{#1} \gdef\groupoidFunction#1{#1} \gdef\groupoidHomomorphism#1{#1} \gdef\affineModifier#1{\overrightharpoon{#1}} \gdef\supercirc#1{#1^{\circ}} \gdef\supercircb#1{#1^{\bullet}} \gdef\blackCircleModifier#1{\supercircb{#1}} \gdef\whiteCircleModifier#1{\supercirc{#1}} \gdef\totalSpaceStyle#1{\piFo{#1}} \gdef\baseSpaceStyle#1{\blFo{#1}} \gdef\fiberSpaceStyle#1{\reFo{#1}} \gdef\totalSpaceElementStyle#1{\daPiFo{#1}} \gdef\baseSpaceElementStyle#1{\daBlFo{#1}} \gdef\fiberSpaceElementStyle#1{\daReFo{#1}} \gdef\bundleProjectionStyle#1{\daGrFo{#1}} \gdef\bundleGraphStyle#1{\piFo{#1}} \gdef\bundleSectionStyle#1{\daOrFo{#1}} \gdef\bundleFunctionStyle#1{\daPiFo{#1}} \gdef\graphFunctionStyle#1{\totalSpaceStyle{#1}} \gdef\projectionFunctionStyle#1{\totalSpaceStyle{#1}} \gdef\sectionFunctionStyle#1{\totalSpaceStyle{#1}} \gdef\functionGraph#1{G_{#1}} \gdef\toroidalModifier#1{\supercirc{#1}} \gdef\modulo#1{\supercirc{#1}} \gdef\dividesSymbol{\mathrel{|}} \gdef\groupFunction#1{#1} \gdef\groupHomomorphism#1{#1} \gdef\automorphisms{\operatorname{Aut}} \gdef\endomorphisms{\operatorname{End}} \gdef\transportMap#1{\transportMapSymbol_{#1}} \gdef\transportMapSymbol{\tau} \gdef\action#1{#1} \gdef\selfAction#1{\hat{#1}} \gdef\actionGroupoid#1{\utilde{#1}} \gdef\cardinalGroup#1{G^*({#1})} \gdef\signed#1{{#1}^*} \gdef\transportAtlas#1{T_{#1}} \gdef\inverted#1{\underline{#1}} \gdef\mirror#1{\overline{#1}} \gdef\pathComposeSymbol{\mathbin{∷}} \gdef\pathCompose#1#2{{#1}\pathComposeSymbol{#2}} \gdef\translateSymbol{\mathbin{\uparrow}} \gdef\backwardTranslateSymbol{\mathbin{\downarrow}} \gdef\pathHead#1{\pathHeadVector{#1}} \gdef\pathTail#1{\pathTailVector{#1}} \gdef\pathHeadVector#1{{#1}^{\bullet}} \gdef\pathTailVector#1{{#1}_{\bullet}} \gdef\pathReverse#1{{#1}^{\dagger}} \gdef\pathIntegral#1#2{{#1} \int {#2}} \gdef\pathIntegralSymbol{{\int}} \gdef\pathDot#1#2{{#1} \cdot {#2}} \gdef\pathDotSymbol{{\cdot}} \gdef\compactBasis#1{\mathscr{B}} \gdef\length{\operatorname{len}} \gdef\signedLength{\operatorname{len^*}} \gdef\andFn{\operatorname{and}} \gdef\orFn{\operatorname{or}} \gdef\notFn{\operatorname{not}} \gdef\vertexList{\operatorname{vertices}} \gdef\vertexList{\operatorname{vertices}} \gdef\edgeList{\operatorname{edges}} \gdef\pathList{\operatorname{paths}} \gdef\cardinalList{\operatorname{cards}} \gdef\signedCardinalList{\operatorname{cards^*}} \gdef\wordOf{\operatorname{word}} \gdef\headVertex{\operatorname{head}} \gdef\tailVertex{\operatorname{tail}} \gdef\basis{\operatorname{basis}} \gdef\split{\operatorname{split}} \gdef\lcm{\operatorname{lcm}} \gdef\minimalContractionSets{\operatorname{MCSets}} \gdef\minimalContractions{\operatorname{MC}} \gdef\grade{\operatorname{grade}} \gdef\support{\operatorname{support}} \gdef\coefficient{\operatorname{coeff}} \gdef\domain{\operatorname{domain}} \gdef\codomain{\operatorname{codomain}} \gdef\modFunction{\operatorname{mod}} \gdef\clip{\operatorname{clip}} \gdef\sign{\operatorname{sign}} \gdef\step{\operatorname{step}} \gdef\projection{\operatorname{proj}} \gdef\lift{\operatorname{lift}} \gdef\identity{\operatorname{id}} \gdef\total{\operatorname{total}} \gdef\torus{\operatorname{torus}} \gdef\mobius{\operatorname{mobius}} \gdef\stateCompose{\operatorname{glue}} \gdef\infixStateComposeSymbol{\_} \gdef\stateDecompose{\operatorname{melt}} \gdef\stateJoin{\operatorname{conj}} \gdef\stateMeet{\operatorname{disj}} \gdef\stateExtent{\operatorname{extent}} \gdef\stateIntent{\operatorname{intent}} \gdef\infixStateJoinSymbol{\sqcup} \gdef\infixStateMeetSymbol{\sqcap} \gdef\isPrime#1{#1\textrm{ prime}} \gdef\blank{\_} \gdef\emptyWord{} \gdef\multiwordSymbol#1{\mathbf{#1}} \gdef\wordSymbol#1{\mathtt{#1}} \gdef\word#1{#1} \gdef\pathMap#1{#1} \gdef\function#1{#1} \gdef\imageModifier#1{{#1}^{\rightarrow}} \gdef\preimageModifier#1{{#1}^{\leftarrow}} \gdef\multiImageColorModifier#1{\msetCol{#1}} \gdef\multiImageModifier#1{{#1}^{\Rightarrow}} \gdef\multiPreimageModifier#1{{#1}^{\Leftarrow}} \gdef\functionComposition#1{#1} \gdef\functionCompositionSymbol{\mathbin{\small ∘}} \gdef\rightFunctionComposition#1{#1} \gdef\rightFunctionCompositionSymbol{\mathbin{\tiny ●}} \gdef\route#1#2#3{[{#1}\!:\!{#2}\!:\!{#3}]} \gdef\multiroute#1#2#3{[{#1}\!:\!{#2}\!:\!{#3}]} \gdef\pathWord#1#2#3{{#1}\!:\!{#2}\!:\!{#3}} \gdef\parenPathWord#1#2#3{\left(\pathWord{#1}{#2}{#3}\right)} \gdef\nullPath{\bot} \gdef\nullElement{\bot} \gdef\path#1{#1} \gdef\vert#1{#1} \gdef\underdot#1{\underset{\raisebox{0.3em}{.}}{#1}} \gdef\headVertexSymbol{◨} \gdef\tailVertexSymbol{◧} \gdef\placeholderVertexSymbol{\mathrlap{◨}{◧}} \gdef\tvert#1{\underdot{#1}} \gdef\hvert#1{\dot{#1}} \gdef\edge#1{#1} \gdef\card#1{\mathtt{#1}} \gdef\path#1{#1} \gdef\quiver#1{#1} \gdef\bindingRuleSymbol{\to} \gdef\compactBindingRuleSymbol{:} \gdef\cayleyQuiverSymbol#1{\selfAction{#1}} \gdef\bindCayleyQuiver#1#2{\selfAction{#1}[#2]} \gdef\bindActionQuiver#1#2{#1[#2]} \gdef\bindSize#1#2{#1(#2)} \gdef\bindCardSize#1#2{#1[#2]} \gdef\bindCards#1#2{#1[#2]} \gdef\subSize#1#2{#1_{#2}} \gdef\gridQuiver#1{\textrm{Grid}^{#1}} \gdef\treeQuiver#1{\textrm{Tree}^{#1}} \gdef\bouquetQuiver#1{\textrm{Bq}^{#1}} \gdef\lineQuiver{\textrm{L}} \gdef\cycleQuiver{\textrm{C}} \gdef\squareQuiver{\textrm{Sq}} \gdef\cubicQuiver{\textrm{Cbc}} \gdef\triangularQuiver{\textrm{Tri}} \gdef\hexagonalQuiver{\textrm{Hex}} \gdef\rhombilleQuiver{\textrm{Rmb}} \gdef\limit#1#2{\lim_{#2}\,#1} \gdef\realVectorSpace#1{\mathbb{R}^{#1}} \gdef\complexVectorSpace#1{\mathbb{C}^{#1}} \gdef\matrixRing#1#2{M_{#2}(#1)} \gdef\groupoid#1{#1} \gdef\group#1{#1} \gdef\field#1{#1} \gdef\ring#1{#1} \gdef\variable#1{#1} \gdef\semiring#1{#1} \gdef\sym#1{#1} \gdef\matrix#1{#1} \gdef\tupleSym#1{#1} \gdef\polynomial#1{#1} \gdef\setLetter{\mathcal{S}} \gdef\signedSetLetter{\mathcal{S^*}} \gdef\multisetLetter{\mathcal{M}} \gdef\signedMultisetLetter{\mathcal{M^*}} \gdef\multisetSemiringSymbol#1{#1} \gdef\multisetSemiring#1#2{\multisetLetter\left[#1, #2\right]} \gdef\signedMultisetRingSymbol#1{#1} \gdef\signedMultisetRing#1#2{\signedMultisetLetter\left[#1, #2\right]} \gdef\polynomialRing#1#2{#1[{#2}]} \gdef\routeSymbol#1{#1} \gdef\multirouteSymbol#1{\mathbf{#1}} \gdef\planSymbol#1{\mathbf{#1}} \gdef\ringElement#1{#1} \gdef\tuplePart#1#2{#1\llbracket{#2}\rrbracket} \gdef\matrixPart#1#2#3{#1\llbracket{#2,#3}\rrbracket} \gdef\matrixRowPart#1{#1} \gdef\matrixColumnPart#1{#1} \gdef\subMatrixPart#1#2#3{#1_{#2,#3}} \gdef\matrixDotSymbol{\cdot} \gdef\matrixPlusSymbol{+} \gdef\wordGroup#1{\wordGroupSymbol_{#1}} \gdef\wordGroupSymbol{\Omega} \gdef\wordRing#1{\wordRingSymbol_{#1}} \gdef\wordRingSymbol{\Omega\!\degree} \gdef\wordRingElement#1{#1} \gdef\wordRingBasisElement#1{e_{#1}} \gdef\linearCombinationCoefficient#1{{\textcolor{888888}{#1}}} \gdef\plan#1{#1} \gdef\planRing#1{\planRingSymbol_{#1}} \gdef\planRingSymbol{\Phi} \gdef\basisPath#1#2{\mathbf{#1}_{#2}} \gdef\basisPathWeight#1#2{{#1}_{#2}} \gdef\unitSymbol{\mathbf{e}} \gdef\unitVertexField{\unitSymbol_1} \gdef\forwardSymbol{f} \gdef\backwardSymbol{b} \gdef\symmetricSymbol{s} \gdef\antisymmetricSymbol{a} \gdef\wordVector#1#2{\unitSymbol_{#1}^{#2}} \gdef\gradOf#1{\grad\,{#1}} \gdef\grad{\nabla} \gdef\divOf#1{\div\,{#1}} \gdef\div{\dot{\nabla}} \gdef\laplacianOf#1{\laplacian\,{#1}} \gdef\laplacian{\ddot{\nabla}} \gdef\suchThat#1#2{{#1}\,\big|\,{#2}} \gdef\chart#1{\chartSymbol_{#1}} \gdef\chartSymbol{C} \gdef\graphRegionIntersectionSymbol{\cap} \gdef\graphRegionUnionSymbol{\cup} \gdef\pathIso{\simeq} \gdef\factorial#1{#1!} \gdef\power#1#2{{#1}^{#2}} \gdef\repeatedPower#1#2{{#1}^{#2}} \gdef\kroneckerDeltaForm#1{\kroneckerDeltaSymbol_{#1}} \gdef\kroneckerDeltaSymbol{\delta} \gdef\contractionLattice#1{\operatorname{Con}(#1)} \gdef\contractedRelation#1{\sim_{#1}} \gdef\isContracted#1#2{{#1} \sim {#2}} \gdef\isContractedIn#1#2#3{{#1} \sim_{#3} {#2}} \gdef\isNotContracted#1#2{{#1} \not \sim {#2}} \gdef\isNotContractedIn#1#2#3{{#1} \not \sim_{#3} {#2}} \gdef\contractionSum#1{#1} \gdef\contractionSumSymbol{\sqcup} \gdef\contractionProduct#1{#1} \gdef\contractionProductSymbol{{\cdot}} \gdef\graph#1{#1} \gdef\graphHomomorphism#1{#1} \gdef\coversSymbol{\sqsupseteq} \gdef\coveredBySymbol{\sqsubseteq} \gdef\strictlyCoversSymbol{\sqsupset} \gdef\strictlyCoveredBySymbol{\sqsubset} \gdef\graphCovering#1#2#3{{#2} \sqsupseteq_{#1} {#3}} \gdef\quiverCovering#1#2#3{{#2} \sqsupseteq^{#1} {#3}} \gdef\powerSetSymbol{\mathcal{P}} \gdef\powerSet#1{\powerSetSymbol({#1})} \gdef\existsForm#1#2{\exists\,{#1}\,:\,{#2}} \gdef\forAllForm#1#2{\forall\,{#1}\,:\,{#2}} \gdef\notted#1{\notSymbol {#1}} \gdef\andSymbol{\land} \gdef\orSymbol{\lor} \gdef\notSymbol{\lnot} \gdef\latticeSymbol#1{#1} \gdef\meetSemilatticeSymbol#1{#1} \gdef\joinSemilatticeSymbol#1{#1} \gdef\posetSymbol#1{#1} \gdef\latticeElementSymbol#1{#1} \gdef\meetSemilatticeElementSymbol#1{#1} \gdef\joinSemilatticeElementSymbol#1{#1} \gdef\posetElementSymbol#1{#1} \gdef\latticeMeetSymbol{\wedge} \gdef\latticeJoinSymbol{\vee} \gdef\latticeTop{\top} \gdef\latticeBottom{\bot} \gdef\semilatticeMeetSymbol{\wedge} \gdef\semilatticeJoinSymbol{\vee} \gdef\semilatticeTop{\top} \gdef\semilatticeBottom{\bot} \gdef\semilatticeSemimeetSymbol{\wedge} \gdef\semilatticeSemijoinSymbol{\vee} \gdef\latticeGreaterSymbol{>} \gdef\latticeGreaterEqualSymbol{\ge} \gdef\latticeLessSymbol{<} \gdef\latticeLessEqualSymbol{\le} \gdef\posetGreaterSymbol{>} \gdef\posetGreaterEqualSymbol{\ge} \gdef\posetLessSymbol{<} \gdef\posetLessEqualSymbol{\le} \gdef\posetCoversSymbol{⋗} \gdef\posetCoveredBySymbol{⋖} \gdef\grpname#1{\operatorname{\mathsf{#1}}} \gdef\mkg#1#2#3{\grpname{#1}({#2},{#3})} \gdef\mka#1#2#3{\mathfrak{#1}({#2},{#3})} \gdef\generalLinearAlgebra#1#2{\mka{gl}{#1}{#2}} \gdef\generalLinearGroup#1#2{\mkg{GL}{#1}{#2}} \gdef\specialLinearAlgebra#1#2{\mka{sl}{#1}{#2}} \gdef\specialLinearGroup#1#2{\mkg{SL}{#1}{#2}} \gdef\projectiveGeneralLinearAlgebra#1#2{\mka{pgl}{#1}{#2}} \gdef\projectiveGeneralLinearGroup#1#2{\mkg{PGL}{#1}{#2}} \gdef\projectiveSpecialLinearAlgebra#1#2{\mka{psl}{#1}{#2}} \gdef\projectiveSpecialLinearGroup#1#2{\mkg{PSL}{#1}{#2}} \gdef\orthogonalAlgebra#1#2{\mka{o}{#1}{#2}} \gdef\orthogonalGroup#1#2{\mkg{O}{#1}{#2}} \gdef\specialOrthogonalAlgebra#1#2{\mka{so}{#1}{#2}} \gdef\specialOrthogonalGroup#1#2{\mkg{SO}{#1}{#2}} \gdef\unitaryAlgebra#1{\mka{u}{#1}{#2}} \gdef\unitaryGroup#1{\mkg{U}{#1}{#2}} \gdef\specialUnitaryAlgebra#1#2{\mka{su}{#1}{#2}} \gdef\specialUnitaryGroup#1#2{\mkg{su}{#1}{#2}} \gdef\spinAlgebra#1#2{\mka{spin}{#1}{#2}} \gdef\spinGroup#1#2{\mkg{Spin}{#1}{#2}} \gdef\pinAlgebra#1#2{\mka{pin}{#1}{#2}} \gdef\pinGroup#1#2{\mkg{Pin}{#1}{#2}} \gdef\symmetricGroup#1{\grpname{Sym}({#1})} \gdef\presentation#1{#1} \gdef\groupPresentation#1#2{\left\langle\,{#1}\,\,\middle|\mathstrut\,\,{#2}\,\right\rangle} \gdef\groupRelationIso{=} \gdef\groupGenerator#1{#1} \gdef\groupRelator#1{#1} \gdef\groupElement#1{#1} \gdef\identityElement#1{#1} \gdef\groupoidElement#1{#1} \gdef\groupIdentity#1{#1} \gdef\groupoidIdentityElement#1#2{#1_{#2}} \gdef\ringIdentity#1{#1} \gdef\iconstruct#1#2{{#1}\,\,\middle|{\large\mathstrut}\,\,{#2}} \gdef\setConstructor#1#2{\left\{\,\iconstruct{#1}{#2}\,\right\}} \gdef\multisetConstructor#1#2{\left\{\mkern{-2.3pt}\left|\,\,\iconstruct{#1}{#2}\,\right|\mkern{-2.3pt}\right\}} \gdef\cardinalityConstructor#1#2{\left|\,\iconstruct{#1}{#2}\,\right|} \gdef\setToMultiset#1{{#1}^\uparrow} \gdef\multisetToSet#1{{#1}^\downarrow} \gdef\subsets#1{\setLetter({#1})} \gdef\signedSubsets#1{\signedSetLetter({#1})} \gdef\multisets#1{\multisetLetter({#1})} \gdef\signedMultisets#1{\signedMultisetLetter({#1})} \gdef\circleSpaceSymbol{S} \gdef\topologicalSpace#1{#1} \gdef\bundleSection#1{#1} \gdef\bundleProjection#1{#1} \gdef\setSymbol#1{#1} \gdef\signedSetSymbol#1{#1} \gdef\multisetSymbol#1{#1} \gdef\signedMultisetSymbol#1{#1} \gdef\setElementSymbol#1{#1} \gdef\signedSetElementSymbol#1{#1} \gdef\multisetElementSymbol#1{#1} \gdef\signedMultisetElementSymbol#1{#1} \gdef\negated#1{\bar{#1}} \gdef\positiveSignedPart#1{{#1}^+} \gdef\negativeSignedPart#1{{#1}^-} \gdef\multisetMultiplicitySymbol{\,\raisebox{.1em}{\small\#}\mkern{.1em}\,} \gdef\signedMultisetMultiplicitySymbol{\,\raisebox{.1em}{\small\#}\mkern{.1em}\,} \gdef\boundMultiplicityFunction#1{{#1}^{\sharp}} \gdef\boundSignedMultiplicityFunction#1{\operatorname{{#1}^{\sharp}}} \gdef\constructor#1#2{\left.{#1}\,\,\middle|\mathstrut\,\,{#2}\right.} \gdef\elemOf#1#2{{ {#1} \in {#2} }} \gdef\notElemOf#1#2{{ {#1} \notin {#2} }} \gdef\edgeOf#1#2{{ {#1} {\in}_E {#2} }} \gdef\vertOf#1#2{{ {#1} {\in}_V {#2} }} \gdef\pathOf#1#2{{ {#1} {\in}_P {#2} }} \gdef\pathType#1{\operatorName{path}{#1}} \gdef\vertexType#1{\operatorName{vertex}{#1}} \gdef\edgeType#1{\operatorName{edge}{#1}} \gdef\multisetType#1{\operatorName{mset}{#1}} \gdef\signedMultisetType#1{\operatorName{mset^*}{#1}} \gdef\submset{\mathbin{\dot{\subset}}} \gdef\submseteq{\mathbin{\dot{\subseteq}}} \gdef\supmset{\mathbin{\dot{\supset}}} \gdef\supmseteq{\mathbin{\dot{\supseteq}}} \gdef\unitInterval{\mathbb{I}} \gdef\fromTo#1{#1} \gdef\fromToSymbol{\mapsto} \gdef\vertexCountOf#1{|{#1}|} \gdef\vertices#1{ V_{#1} } \gdef\edges#1{ E_{#1} } \gdef\vertexField#1{#1} \gdef\edgeField#1{#1} \gdef\pathVector#1{\mathbf{#1}} \gdef\pathVectorSpace#1{\mathscr{#1}} \gdef\baseField#1{#1} \gdef\finiteField#1{\mathbb{F}_{#1}} \gdef\functionSpace#1#2{#2^{#1}} \gdef\finiteTotalFunctionSpace#1#2{#2^{\sub #1}} \gdef\pathGroupoid#1{{ \Gamma_{#1} }} \gdef\forwardPathQuiver#1#2{{\overrightharpoon{#1}_{#2}}} \gdef\backwardPathQuiver#1#2{{\overrightharpoon{#1}^{#2}}} \gdef\pathQuiver#1{{\overrightharpoon{#1}}} \gdef\mto#1#2{{#1}\mapsto{#2}} \gdef\mtoSymbol{\mapsto} \gdef\groupWordRewriting#1{\langle{#1}\rangle} \gdef\rewrite#1#2{{#1}\mapsto{#2}} \gdef\rewritingRule#1#2{{#1}\mapsto{#2}} \gdef\rewritingSystem#1{\mathcal{#1}} \gdef\multiwayBFS#1{\textrm{bfs}({#1})} \gdef\rewritingStateBinding#1#2{{\left.{#1}\!\mid\!{#2}\right.}} \gdef\rewritingRuleBinding#1#2{#1{\left[\,{#2}\,\right]}} \gdef\namedSystem#1{\mathtt{#1}} \gdef\genericRewritingSystem{\namedSystem{Sys}} \gdef\stringRewritingSystem{\namedSystem{Str}} \gdef\circularStringRewritingSystem{\namedSystem{CStr}} \gdef\turingMachineRewritingSystem{\namedSystem{TM}} \gdef\cellularAutomatonRewritingSystem{\namedSystem{CA}} \gdef\graphRewritingSystem{\namedSystem{Gr}} \gdef\hypergraphRewritingSystem{\namedSystem{HGr}} \gdef\petriNetRewritingSystem{\namedSystem{Petri}} \gdef\localStates#1{\localStatesSymbol^#1} \gdef\regionalStates#1{\regionalStatesSymbol^#1} \gdef\globalStates#1{\globalStatesSymbol^#1} \gdef\keySubStates#1{\keySubStatesSymbol^#1} \gdef\valueSubStates#1{\valueSubStatesSymbol^#1} \gdef\localState#1#2{#2_{#1}} \gdef\regionalState#1{#1} \gdef\globalState#1{#1} \gdef\keySubState#1{#1} \gdef\valueSubState#1{#1} \gdef\lhsState#1{#1_{L}} \gdef\rhsState#1{#1_{R}} \gdef\rewriteLHSRegionalState#1{#1_{L}} \gdef\rewriteRHSRegionalState#1{#1_{R}} \gdef\regionalStateForm#1{\daGFo{(}#1\daGFo{)}} \gdef\invalidRegionalState{\reFo{\times}} \gdef\emptyRegionalState{\regionalStateForm{}} \gdef\regionalSubstateSymbol{\sqsubseteq} \gdef\regionalSuperstateSymbol{\sqsupseteq} \gdef\comparableRegionalStatesSymbol{\mathbin{\square}} \gdef\incomparableRegionalStatesSymbol{\mathbin{\boxtimes}} \gdef\namedStateSet#1{\mathbf #1} \gdef\localStatesSymbol{\namedStateSet L} \gdef\regionalStatesSymbol{\namedStateSet R} \gdef\globalStatesSymbol{\namedStateSet G} \gdef\keySubStatesSymbol{\namedStateSet L_K} \gdef\valueSubStatesSymbol{\namedStateSet L_V} \gdef\localStateSymbol#1{#1} \gdef\regionalStateSymbol#1{#1} \gdef\globalStateSymbol#1{#1} \gdef\keySubStateSymbol#1{#1} \gdef\valueSubStateSymbol#1{#1} \gdef\infixComposeLocalStatesSymbol{\_} \gdef\composeLocalStatesSymbol{\operatorname{glue}} \gdef\composeLocalStatesForm#1{\composeLocalStatesSymbol(#1)} \gdef\ncard#1{\card{\inverted{#1}}} \gdef\mcard#1{\card{\mirror{#1}}} \gdef\nmcard#1{\card{\inverted{\mirror{#1}}}} \gdef\assocArray#1{\left\langle {#1} \right\rangle} \gdef\openMultiset{\lBrace} \gdef\closeMultiset{\rBrace} \gdef\set#1{\{ {#1} \}} \gdef\signedSet#1{\{ {#1} \}} \gdef\list#1{\{ {#1} \}} \gdef\multiset#1{\openMultiset {#1} \closeMultiset} \gdef\signedMultiset#1{\openMultiset {#1} \closeMultiset} \gdef\styledSet#1#2{#1\{ {#1} #1\}} \gdef\styledList#1#2{#1\{ {#1} #1\}} \gdef\styledMultiset#1#2{#1\openMultiset {#2} #1\closeMultiset} \gdef\styledSignedMultiset#1#2{\openMultiset {#2} \closeMultiset} \gdef\permutationCycle#1{#1} \gdef\permutationCycleSymbol{\to} \gdef\permutationSet#1{#1} \gdef\permutationSetSymbol{;} \gdef\transposition#1#2{{#1} \leftrightarrow {#2}} \gdef\tuple#1{( {#1} )} \gdef\concat#1{ {#1} } \gdef\paren#1{\left( {#1} \right)} \gdef\ceiling#1{\lceil{#1}\rceil} \gdef\floor#1{\lfloor{#1}\rfloor} \gdef\translationVector#1{\left[{#1}\right]_{\textrm{T}}} \gdef\quotient#1#2{{#1} / {#2}} \gdef\compactQuotient#1#2#3{{#1}\;{\upharpoonright_{#2}}\;{#3}} \gdef\multilineQuotient#1#2{\frac{#1}{#2}} \gdef\idElem#1{e_{#1}} \gdef\minus#1{-{#1}} \gdef\elem{\ \in\ } \gdef\vsp{\mkern{0.4pt}} \gdef\iGmult{\vsp} \gdef\gmult{\vsp\ast\vsp} \gdef\Gmult{\vsp\ast\vsp} \gdef\gdot{\vsp\cdot\vsp} \gdef\gDot{\vsp\mathbin{\largeDot}\vsp} \gdef\mdot{\vsp\cdot\vsp} \gdef\smallblackcirc{\vsp\raisebox{0.15em}{\tiny∙}\vsp} \gdef\smallwhitecirc{\vsp\raisebox{0.15em}{\tiny∘}\vsp} \gdef\sgdot{\mathbin{\smallwhitecirc}} \gdef\srdot{\mathbin{\smallblackcirc}} \gdef\srplus{+} \gdef\verticalEllipsis{\vdots} \gdef\appliedRelation#1{#1} \gdef\setUnionSymbol{\cup} \gdef\setIntersectionSymbol{\cap} \gdef\setRelativeComplementSymbol{-} \gdef\msetCol{\textcolor{bb4444}} \gdef\repeatedMultiset#1#2{#1\,#2} \gdef\msrdot{\mathbin{\smallblackcirc}} \gdef\msrplus{+} \gdef\smrdot{\mathbin{\smallblackcirc}} \gdef\smrplus{+} \gdef\dotminus{\mathbin{\dot{-}}} \gdef\dotcap{\mathbin{\dot{\cap}}} \gdef\dotcup{\mathbin{\dot{\cup}}} \gdef\multisetUnionSymbol{\dotcup} \gdef\multisetIntersectionSymbol{\dotcap} \gdef\multisetRelativeComplementSymbol{\dotminus} \gdef\multisetSumSymbol{\dotplus} \gdef\cartesianProductSymbol{\times} \gdef\functionType#1#2{{#1} \to {#2}} \gdef\functionSignature#1#2#3{{{#1} : {#2} \to {#3}}} \gdef\partialFunctionSignature#1#2#3{{{#1} : {#2} \rightharpoonup {#3}}} \gdef\poly#1{#1} \gdef\quiverProdPoly#1{#1} \gdef\quiverProdPower#1#2{#1^{#2}} \gdef\quiverProdTimes#1{#1} \gdef\parenLabeled#1#2{#1\ ({#2})} \gdef\parenRepeated#1#2{\parenLabeled{#1}{{#2}\text{ times}}} \gdef\underLabeled#1#2{\underbrace{#1}_{#2}} \gdef\underRepeated#1#2{\underbrace{#1}_{#2\text{ times}}} \gdef\overLabeled#1#2{\overbrace{#1}^{#2}} \gdef\overRepeated#1#2{\overbrace{#1}^{#2\text{ times}}} \gdef\modLabeled#1#2{{#1 }\textrm{ mod }{#2}} \gdef\freeGroup#1{F_{#1}} \gdef\cyclicGroup#1{\mathbb{Z}_{#1}} \gdef\componentSuperQuiverOfSymbol{\succ} \gdef\setCardinality#1{|{#1}|} \gdef\multisetCardinality#1{|{#1}|} \gdef\dependentQuiverProductSymbol{\mathbin{\times}} \gdef\rightIndependentQuiverProductSymbol{\mathbin{⋊}} \gdef\leftIndependentQuiverProductSymbol{\mathbin{⋉}} \gdef\rightStrongQuiverProductSymbol{\mathbin{⧒}} \gdef\leftStrongQuiverProductSymbol{\mathbin{⧑}} \gdef\rightFiberQuiverProductSymbol{\mathbin{⧕}} \gdef\leftFiberQuiverProductSymbol{\mathbin{⧔}} \gdef\lockedQuiverProductSymbol{\mathbin{\searrow}} \gdef\rightFreeQuiverProductSymbol{\mathbin{\smallerthan}} \gdef\leftFreeQuiverProductSymbol{\mathbin{\largerthan}} \gdef\strongIndependentQuiverProductSymbol{\mathbin{⨝}} \gdef\cartesianQuiverProductSymbol{\mathbin{□}} \gdef\strongQuiverProductSymbol{\mathbin{⊠}} \gdef\graphUnionSymbol{\sqcup} \gdef\graphProductSymbol{\times} \gdef\inlineProdSymbol{|} \gdef\serialCardSymbol{{:}} \gdef\parallelCardSymbol{{\mid}} \gdef\cardinalSequenceSymbol{{:}} \gdef\cardinalProduct#1{(#1)} \gdef\vertexProduct#1{(#1)} \gdef\edgeProduct#1{(#1)} \gdef\cardinalProductSymbol{\inlineProdSymbol} \gdef\vertexProductSymbol{\inlineProdSymbol} \gdef\edgeProductSymbol{\inlineProdSymbol} \gdef\verticalVertexProduct#1#2{\cfrac{#1}{#2}} \gdef\verticalCardinalProduct#1#2{\cfrac{#1}{#2}} \gdef\indexSum#1#2#3{{\sum_{#2}^{#3} #1}} \gdef\indexProd#1#2#3{{\prod_{#2}^{#3} #1}} \gdef\indexMax#1#2#3{{\max_{#2}^{#3} #1}} \gdef\indexMin#1#2#3{{\min_{#2}^{#3} #1}} \gdef\indexUnion#1#2#3{{\bigcup_{#2}^{#3} #1}} \gdef\indexIntersection#1#2#3{{\bigcap_{#2}^{#3} #1}} \gdef\indexGraphUnion#1#2#3{{\bigcup_{#2}^{#3} #1}} \gdef\indexGraphDisjointUnion#1#2#3{{\bigcup_{#2}^{#3} #1}} \gdef\styledIndexSum#1#2#3#4{{#1\sum_{#3}^{#4} #2}} \gdef\styledIndexProd#1#2#3#4{{#1\prod_{#3}^{#4} #2}} \gdef\styledIndexMax#1#2#3#4{{#1\max_{#3}^{#4} #2}} \gdef\styledIndexMin#1#2#3#4{{#1\min_{#3}^{#4} #2}} \gdef\indexSumSymbol{\sum} \gdef\indexProdSymbol{\prod} \gdef\indexMaxSymbol{\max} \gdef\indexMinSymbol{\min} \gdef\openInterval#1#2{(#1,#2)} \gdef\closedInterval#1#2{[#1,#2]} \gdef\openClosedInterval#1#2{(#1,#2]} \gdef\closedOpenInterval#1#2{[#1,#2)} \gdef\oneTo#1{1..{#1}} \gdef\zeroTo#1{0..{#1}} \gdef\qstring#1{\mathtt{"}{#1}\mathtt{"}} \gdef\wstring#1{\textcolor{6b6b6b}{#1}} \gdef\qchar#1{\mathtt{'}{#1}\mathtt{'}} \gdef\lstr#1{\mathtt{#1}} \gdef\lchar#1{\mathtt{#1}} \gdef\string#1{{#1}} \gdef\character#1{{#1}} \gdef\homomorphismMapping#1{\assocArray{#1}} \gdef\starModifier#1{{#1}^*} \gdef\translationPresentation#1{\textrm{Z}_{#1}} \gdef\starTranslationPresentation#1{\textrm{Z}^*_{#1}} \gdef\translationPathValuation#1{\mathcal{\overrightharpoon Z}_{#1}} \gdef\starTranslationPathValuation#1{\overrightharpoon{\mathcal{Z}^*_{#1}}} \gdef\translationWordHomomorphism#1{\mathcal{Z}_{#1}} \gdef\starTranslationWordHomomorphism#1{\mathcal{Z}^*_{#1}} \gdef\translationCardinalValuation#1{\textrm{T}_{#1}} \gdef\starTranslationCardinalValuation#1{\textrm{T}^*_{#1}} \gdef\reFo#1{\textcolor{e1432d}{#1}} \gdef\grFo#1{\textcolor{4ea82a}{#1}} \gdef\blFo#1{\textcolor{3e81c3}{#1}} \gdef\orFo#1{\textcolor{dc841a}{#1}} \gdef\piFo#1{\textcolor{c74883}{#1}} \gdef\teFo#1{\textcolor{47a5a7}{#1}} \gdef\gFo#1{\textcolor{929292}{#1}} \gdef\puFo#1{\textcolor{8b7ebe}{#1}} \gdef\liReFo#1{\textcolor{ff775e}{#1}} \gdef\liGrFo#1{\textcolor{82dd63}{#1}} \gdef\liBlFo#1{\textcolor{6caff4}{#1}} \gdef\liOrFo#1{\textcolor{ffbb5f}{#1}} \gdef\liPiFo#1{\textcolor{fb77b0}{#1}} \gdef\liTeFo#1{\textcolor{7fdbdc}{#1}} \gdef\liGFo#1{\textcolor{c5c5c5}{#1}} \gdef\liPuFo#1{\textcolor{bbaff2}{#1}} \gdef\daReFo#1{\textcolor{b50700}{#1}} \gdef\daGrFo#1{\textcolor{217f00}{#1}} \gdef\daBlFo#1{\textcolor{165e9d}{#1}} \gdef\daOrFo#1{\textcolor{ae5900}{#1}} \gdef\daPiFo#1{\textcolor{9e1f61}{#1}} \gdef\daTeFo#1{\textcolor{0e7c7e}{#1}} \gdef\daGFo#1{\textcolor{6b6b6b}{#1}} \gdef\daPuFo#1{\textcolor{665996}{#1}} \gdef\boFo#1{\mathbf{#1}} \gdef\itFo#1{\mathit{#1}} \gdef\unFo#1{\underline{#1}} \gdef\stFo#1{\struckthrough{#1}} \gdef\plTeFo#1{\textrm{#1}} \gdef\maTeFo#1{\textrm{#1}} \gdef\ZNFuFo#1{\operatorname{#1}} \gdef\ZAFo#1#2{#1(#2)} \gdef\noApSy{\gFo{\text{---}}} \gdef\unSy{\gFo{?}} \gdef\emSeSy{\gFo{ \emptyset }} \gdef\tiSy{\boFo{ \vthinspace ✓ \vthinspace }} \gdef\unIn{\mathbb{I}} \gdef\blSy{\gFo{\_}} \gdef\plSqSy{□} \gdef\baToSy{\boFo{ \vthinspace | \vthinspace }} \gdef\fiToSy{●} \gdef\fiSqToSy{■} \gdef\fiReToSy{▮} \gdef\emToSy{○} \gdef\emSqToSy{□} \gdef\emReToSy{▯} \gdef\plElSy{\,...\,} \gdef\ceElSy{\,\cdot\!\cdot\!\cdot\,} \gdef\elSy{\,\gFo{...}\,} \gdef\veElSy{\vdots} \gdef\coFo{\sqsubseteq} \gdef\CoFo#1#2{#1 \coFo #2} \gdef\coByFo{⊑} \gdef\CoByFo#1#2{#1 \coByFo #2} \gdef\stCoFo{\sqsupset} \gdef\StCoFo#1#2{#1 \stCoFo #2} \gdef\stCoByFo{\sqsubset} \gdef\StCoByFo#1#2{#1 \stCoByFo #2} \gdef\inCoFo{\sqsubseteq} \gdef\InCoFo#1#2{#1 \inCoFo #2} \gdef\InCoFoⅡ#1#2#3{#1 \inCoFo_{#3} #2} \gdef\grInCoFo{\sqsubseteq} \gdef\GrInCoFo#1#2{#1 \grInCoFo #2} \gdef\GrInCoFoⅡ#1#2#3{#1 \grInCoFo_{#3} #2} \gdef\quInCoFo{\sqsubseteq} \gdef\QuInCoFo#1#2{#1 \quInCoFo #2} \gdef\QuInCoFoⅡ#1#2#3{#1 \quInCoFo_{#3} #2} \gdef\coPrFo{\cdot} \gdef\CoPrFo#1{#1} \gdef\coSuFo{\sqcup} \gdef\CoSuFo#1{#1} \gdef\isCoFo{\sim} \gdef\IsCoFo#1#2{#1 \isCoFo #2} \gdef\IsCoFoⅡ#1#2#3{#1 \isCoFo_{#3} #2} \gdef\isNoCoFo{\nsim} \gdef\IsNoCoFo#1#2{#1 \isNoCoFo #2} \gdef\IsNoCoFoⅡ#1#2#3{#1 \isNoCoFo_{#3} #2} \gdef\coReFo{\sim} \gdef\CoReFo#1{#1} \gdef\TuFo#1{(#1)} \gdef\SeFo#1{{#1}} \gdef\StSeFo#1#2#3{#1#3#2} \gdef\inArSy{⏵} \gdef\arSy{⏴} \gdef\upArSy{⏶} \gdef\doArSy{⏷} \gdef\leArSy{⏴} \gdef\riArSy{⏵} \gdef\grAuFu{\operatorname{Aut}} \gdef\PaFo#1{\;#1\;} \gdef\na{\mathbb{n}} \gdef\poNa{\mathbb{N} ^+} \gdef\poRe{\mathbb{R} ^+} \gdef\pr{\mathbb{P}} \gdef\re{\mathbb{R}} \gdef\piSy{\pi} \gdef\taSy{\tau}$

## Graphs and paths #

This is an example of a basic graph:

We've labeled the vertices $$\gdef\ChEmGrPaFo#1{#1^●} \gdef\EmGrPaFo#1{#1:#1} \gdef\LaCoPlSq#1{\textcolor{#1}{\raisebox{-0.1em}{\;▢\;}}} \gdef\fiPlSq{\raisebox{-0.05em}{\mkern{2pt}▧}} \vert{a},\vert{b},\vert{c}$$.

Basic graphs do not have any notion of direction on their edges, and do not have more than one edge between any two vertices.

We write the edges explicitly as $$\ue{\vert{a}}{\vert{b}}$$ etc. We can also refer to them symbolically as $$\edge{e},\edge{f}$$, etc.

Because edges are undirected, $$\ue{\vert{a}}{\vert{b}}\syntaxEqualSymbol \ue{\vert{b}}{\vert{a}}$$

A path on a basic graph is an alternating sequence of vertices and edges, beginning and ending with a vertex, such that each edge matches the neighboring two vertices:

$\vert{a}:\ue{\vert{a}}{\vert{b}}:\vert{b}:\ue{\vert{b}}{\vert{c}}:\edge{c}:\elSy:\edge{z}$

Using symbols for the edges, we can write this as:

$\vert{a}:\edge{e}:\edge{f}:\elSy:\edge{z}$

Alternatively we can write this more compactly by chaining together several edges, where we use arrow-like notation:

$\vert{a} \desym \vert{b} \desym \vert{c} \desym \elSy \desym \vert{z}$

While edges in a basic graph are not oriented, paths necessarily are, since they choose a source vertex and a target vertex, and connect them with edges. Hence chained notation uses an arrow-like form, since we can make two distinct paths out of a single edge:

\begin{aligned} \vert{a} \desym \vert{b}&\neq \vert{b} \desym \vert{a}\\ \ue{\vert{a}}{\vert{b}}&= \ue{\vert{b}}{\vert{a}}\end{aligned}

A path containing no edges must have equal source and target vertices, and so while we can write it fully as $$\EmGrPaFo{\vert{a}}$$, we also can use the more compact notation $$\ChEmGrPaFo{\vert{a}}$$

#### Notation #

A $$\sym{n}$$-path is a path in which $$\sym{n}$$ edges appear.

A 0-path is also called an empty path.

#### Reversal #

We can invert a path by reversing the alternating sequence.

### Chaining of paths #

We two paths are target-to-source, we can chain them together into a single path. We write this:

$\pathCompose{\paren{\vert{a} \desym \vert{b} \desym \vert{c}}}{\paren{\vert{c} \desym \vert{d}}} = \vert{a} \desym \vert{b} \desym \vert{c} \desym \vert{d}$

### Cancellation of paths #

We can adopt the perspective that $$\vert{a} \desym \vert{b} \desym \vert{a} = \ChEmGrPaFo{\vert{a}}$$, in other words that a backtracked pair can be removed from any path without changing its value:

$\PaFo{\PaFo{\reFo{\fiPlSq{}} \desym \vert{a} \desym \vert{b} \desym \vert{a} \desym \blFo{\fiPlSq{}}}}\syntaxEqualSymbol \boFo{\PaFo{\PaFo{\reFo{\fiPlSq{}} \desym \blFo{\fiPlSq{}}}}}$

We illustrate this here on an example:

## Path groupoid #

Chaining of paths in a graph $$\graph{G}$$ forms the operation of a groupoid, the path monoid. When we adopt the perspective of path cancellation, then chaining of paths forms a groupoid, the path groupoid $$\pathGroupoid{\quiver{G}}$$.