Brussels Graph

Author

Eric W. Weisstein
November 7, 2025

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/BrusselsGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BrusselsGraph.html.

©2025 Wolfram Research, Inc. except for portions noted otherwise

Figures

LCF

lcf=GraphData["BrusselsGraph","Graph","LCF"];

(LCFNotation/@lcf)[[All,-1]]//Tally

GraphicsGrid[Partition[lcf,UpTo[6]]//StyleGraphs[#,EdgeStyle->LightDarkSwitched@Black]&,ImageSize->700,Dividers->All]

UnitDistance refinement

GraphData["BrusselsGraph","Graph","UnitDistance"]

udrefinements=Complement@@(GraphData["BrusselsGraph","Graph",#]&/@{"All","LCF"})

uddata=SortBy[{#,Max[Abs[EdgeLengths[#]-1]]}&/@udrefinements,Last]

GraphicsGrid[Partition[Show[StyleGraphs[#],PlotLabel->Style[Row[{Subscript[Row[{"Δ","e"}],"max"],"==",NumberForm[#2,{3,2}]}," "],Directive[FontFamily->"Times"]]]&@@@uddata,
UpTo[4]],
Dividers->All,Alignment->Top,ImageSize->700]

Embeddings

<<MathWorld`Graphs`

Names

GraphData["BrusselsGraph","StandardName"]

GraphData["BrusselsGraph","StandardNames"]

HouseOfGraphs

hogid=GraphData["BrusselsGraph","NotationRules","HouseOfGraphs"]

HouseOfGraphsLookup[hogid]

HouseOfGraphsGraph[hogid]//StyleGraphs[#,ImageSize->Small]&

Primary

GraphData["BrusselsGraph"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["BrusselsGraph","Graph","All"]//StyleGraphs

CurvedEdge

GraphData["BrusselsGraph","Graph","CurvedEdge"]//StyleGraphs

Inexact

Select[GraphData["BrusselsGraph","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Degenerate

GraphData["BrusselsGraph","Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData["BrusselsGraph","Graph","Degenerate"]//Column

Embeddings

GraphDataEmbeddings[GraphData["BrusselsGraph"]]

Default embedding

GraphPlot[Graph[GraphData["BrusselsGraph","VertexList"],GraphData["BrusselsGraph","Edges"]],GraphLayout->"HighDimensionalEmbedding"]

v={};
g=Graph[Range@Length@v,UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

g=FindCorrespondingGraph[,v]

RecognizeGraph@g

GeneralizedPetersen

GraphData["BrusselsGraph","Graph","GeneralizedPetersen"]//StyleGraphs

IGraph

GraphData["BrusselsGraph","Graph","IGraph"]//StyleGraphs

Bipartite embedding

GraphData["BrusselsGraph","Bipartite"]

g=Graph[GraphData["BrusselsGraph","VertexList"],GraphData["BrusselsGraph","Edges"],GraphLayout->"BipartiteEmbedding"]

Bilateral

GraphData["BrusselsGraph","Graph","Bilateral"]//StyleGraphs

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingTypes[#],GraphSymmetryGroup[#]}&/@GraphData["BrusselsGraph","Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]//Quiet

Circular

GraphData["BrusselsGraph","Graph","Circular"]//StyleGraphs

Circular construction

GraphData["BrusselsGraph","Hamiltonian"]

GraphData["BrusselsGraph","LCF"]

GraphData["BrusselsGraph","HamiltonianCycleCount"]

Planar

GraphData["BrusselsGraph","Planar"]

GraphData["BrusselsGraph","Graph","Planar"]//StyleGraphs

MinimalCrossingNumber: "EmbeddingScheme" -> 1

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->1,TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->1,TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->1,TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->1,TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->1,TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->1,TimeConstraint->86400])//AbsoluteTiming

MinimalCrossingNumber: "EmbeddingScheme" -> 2

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->2,TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->2,TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->2,TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->2,TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->2,TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->2,TimeConstraint->86400])//AbsoluteTiming

MinimalCrossingNumber: "EmbeddingScheme" -> 3

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->3,TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->3,TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->3,TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->3,TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->3,TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BrusselsGraph"],"EmbeddingScheme"->3,TimeConstraint->86400])//AbsoluteTiming

MinimalCrossingNumber crossings.uos.de

https://crossings.uos.de/job/new

ExportString[GraphData["BrusselsGraph"],"DOT"]

MinimalCrossing

GraphData["BrusselsGraph","CN"]

(gs=GraphData["BrusselsGraph","Graph","MinimalCrossing"])//StyleGraphs

GraphCrossingCount/@gs

GraphCrossingDiagram/@gs

MinimalRectilinearCrossing

GraphData["BrusselsGraph","RCN"]

GraphData["BrusselsGraph","Graph","MinimalRectilinearCrossing"]//StyleGraphs

MinimalCrossing construction Tutte

(cn=CrossingNumber[GraphData["BrusselsGraph"],TimeConstraint->1])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData["BrusselsGraph"],"CrossingNumber"->cn])//PolyhedralGraphQ//AbsoluteTiming

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

TutteGraphs=BilateralGraph[CenterGraph@IGLayoutTutte[g,"OuterFace"->#],"ReturnNonbilateralGraph"->True]&/@faces

TutteGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["BrusselsGraph","V"]])]
}&/@TutteGraphs;

Cases[TutteGraphsWithOuterCrossingPointCounts,{g_,0}:>g]

Select[Cases[TutteGraphsWithOuterCrossingPointCounts,{g_,1}:>g],Length[OuterFace[#]]>3&]

TutteGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[
#,
AdditionalData->Join[Table[True,GraphData["BrusselsGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@TutteGraphs;

DeleteCases[
TutteGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]

MinimalCrossing construction Random

(cn=CrossingNumber[GraphData["BrusselsGraph"],TimeConstraint->1])//AbsoluteTiming

(RandomGraphs=Table[MinimalCrossingGraph[GraphData["BrusselsGraph"],"CrossingNumber"->cn,RandomSeed->n],{n,1000}]//Union)//Timing

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["BrusselsGraph","V"]])]
}&/@RandomGraphs;

Small numbers of crossing points on the boundary:

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,0}:>g]

Select[Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,1}:>g],Length[OuterFace[#]]>3&]

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[
#,
AdditionalData->Join[Table[True,GraphData["BrusselsGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@RandomGraphs;

DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]/.g_Graph:>BilateralGraph[CenterGraph@g,"ReturnNonbilateralGraph"->True]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

RectilinearLocalCrossingCount[g]

MinimalLocalCrossing

GraphData["BrusselsGraph",{"LocalCrossingNumber","RectilinearLocalCrossingNumber"}]

GraphData["BrusselsGraph","Graph","MinimalLocalCrossing"]//StyleGraphs

MinimalRectilinearLocalCrossing

GraphData["BrusselsGraph",{"LocalCrossingNumber","RectilinearLocalCrossingNumber"}]

GraphData["BrusselsGraph","Graph","MinimalRectilinearLocalCrossing"]//StyleGraphs

MinimaRectilinearlLocalCrossing construction

AdjacentCrossingsQ[g_Graph,cn_Integer]:=Module[{a=AdjacencyList[g],n=VertexCount[g]-cn},Max[Take[a,-cn]]>n]

(cn=CrossingNumber[GraphData["BrusselsGraph"],TimeConstraint->10])//AbsoluteTiming

(gs=Select[Table[MinimalCrossingGraph[GraphData["BrusselsGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData["BrusselsGraph","V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

(gs=Select[Table[MinimalCrossingGraph[GraphData["BrusselsGraph"],"CrossingNumber"->(cn=XXX),RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData["BrusselsGraph","V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

Circulant

GraphData["BrusselsGraph","Circulant"]

GraphData["BrusselsGraph","Graph","Circulant"]//StyleGraphs

Circulant construction

GraphData["BrusselsGraph","NotationRules","Circulant"]

(ci=CirculantIndices[GraphData["BrusselsGraph"]])//Timing

CirculantGraph1@@@ci

Cases[GraphData["BrusselsGraph","StandardNames"],{"Circulant",i_}:>i]

CirculantGraph1@@@%

Complement graph

compl=GraphData["BrusselsGraph","ComplementGraph","Name"]

Line graph

GraphData["BrusselsGraph","Line"]

LineGraphQ[GraphData["BrusselsGraph"]]//Timing

RootGraph

roots=Select[GraphData[All],GraphData[#,"LineGraph","Name"]==="BrusselsGraph"&]

RootGraph[GraphData["BrusselsGraph"]]

Halin

GraphData["BrusselsGraph","Halin"]

GraphData["BrusselsGraph","Graph","Halin"]//StyleGraphs

Rigid

GraphData["BrusselsGraph","Rigid"]

GraphData["BrusselsGraph","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["BrusselsGraph","Flexible"]

GraphData["BrusselsGraph","Graph","Flexible"]//StyleGraphs

LCF

GraphData["BrusselsGraph","LCF"]

GraphData["BrusselsGraph","HamiltonianCycleCount"]

GraphData["BrusselsGraph","LCFSignature"]

GraphData["BrusselsGraph","LCFNotations"]

GraphData["BrusselsGraph","Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute random

FindRandomLCFNotation[GraphData["BrusselsGraph"],TimeConstraint->120]

LCF compute 10^4 [0.1 GB]

Tally[(lcf=LCF[,10^4])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

LCF compute 10^5 [0.35 GB]

Tally[(lcf=LCF[,10^5])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

LCF compute 10^6 [2.7 GB]

Tally[(lcf=LCF[,10^6])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

LCF compute 10^7 [24.8 GB]

Tally[(lcf=LCF[,10^7])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,3,2])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

LCF compute 5*^7 [XX GB]

Tally[(lcf=LCF[,5*^7])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,3,2])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

LCF compute all

Divisors[GraphData["BrusselsGraph","V"]]

Length[FindHamiltonianCycle[GraphData["BrusselsGraph"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["BrusselsGraph"],All])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

UnitDistance

GraphData["BrusselsGraph","UnitDistance"]

UnitDistanceGraphQ[GraphData["BrusselsGraph"],Debug->True]//Timing

GraphData["BrusselsGraph","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement of default embeddings

Graph[edge list]

(g=UnitDistanceEmbeddingRefine[Graph[GraphData["BrusselsGraph","EdgeList"]],MaxIterations->500])//Timing

UnitDistanceGraphEmbeddingQ[g]

Graph[vertex list, edge list]

(g=UnitDistanceEmbeddingRefine[Graph[GraphData["BrusselsGraph","VertexList"],GraphData["BrusselsGraph","EdgeList"]],MaxIterations->500])//Timing

UnitDistanceGraphEmbeddingQ[g]

UnitDistance by refinement of existing embeddings

gs=UnitDistanceEmbeddingRefine[#,MaxIterations->500]&/@Complement@@(GraphData["BrusselsGraph","Graph",#]&/@{"All","UnitDistance"})

TextGrid[
gdata=SortBy[
{#,DegenerateGraphEmbeddingQ[#],GraphSymmetryGroup[#],Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
],
Dividers->All
]

Cases[gdata,{__,0.}]

Symmetric:

gs=Cases[
gdata,
{g_,_,sym:Except[TransformationGroup[{}]],diff_}:>{
HighlightGraph[g1=BilateralGraph[CenterGraph[g],"ReturnNonbilateralGraph"->True],Flatten[FindCoincidentVertices[g1]]],
sym,
diff
}]

UnitDistance by refinement of stored LCF embeddings

FileByteCount[file="/Volumes/GraphProperties/Properties/LCFNotations/"<>GraphNameToFileName["BrusselsGraph"]<>".gz"]

Tally[(lcf=Import[file])[[All,-1]]]//Timing//AbsoluteTiming

lcf=LCFNotation/@{,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,};

gs=UnitDistanceEmbeddingRefine[LCFGraph@@#,MaxIterations->500]&/@Take[lcf,All];//Timing

gdata=SortBy[
{#,Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
]

Cases[gdata,{g_?UnitDistanceGraphEmbeddingQ,_}:>g]

NondegenerateSymmetricGraphs=Select[
gdata,
And[
DegenerateGraphEmbeddingQ[#[[1]]]===False,
GraphSymmetryGroup[#[[1]]]=!=TransformationGroup[{}]
]&
]/.g_Graph:>CenterGraph[g]

NondegenerateSymmetricGraphs[[All,1]]

gs={,,,,,,,};

UnitDistance by FindUnitDistanceEmbedding  (random embeddings returning on messages)

gs=FindUnitDistanceEmbedding[GraphData["BrusselsGraph"],50,"ReturnWithMessage"->True];//Quiet//Timing

gdata=SortBy[
{#,Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
]

Cases[gdata,{g_?UnitDistanceGraphEmbeddingQ,_}:>g]

SymmetricEmebddings=Select[gdata,GraphSymmetryGroup[#[[1]]]=!=TransformationGroup[{}]&]

UnitDistance by FindUnitDistanceEmbedding  (random embedding waiting until UD is found)

FindUnitDistanceEmbedding[GraphData["BrusselsGraph"],"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{$Version,DateString[]}

{"BrusselsGraph",l=FindSubgraphUnitDistanceEmbedding[GraphData["BrusselsGraph"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Select[IndexGraph/@gs,!DegenerateGraphEmbeddingQ[#]&&GraphSymmetryGroup[#]=!=TransformationGroup[{}]&]//Timing

Matchstick

GraphData["BrusselsGraph","Matchstick"]

GraphData["BrusselsGraph","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["BrusselsGraph","Edges"]]

3D construction

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

3D All

GraphData["BrusselsGraph","Graph",{"3D","All"}]

3D UnitDistance

GraphData["BrusselsGraph","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["BrusselsGraph","Polyhedral"]

GraphData["BrusselsGraph","Graph",{"3D","Polyhedron"}]

Construction

van Dam and Haemers 2003 (Wells cospectral graph by switching a pair of antipodal edges)

"There are three distinct graphs having the spectrum of the Wells graph (van Dam and Haemers 2002)."

One is in GraphData as of V15 (2025-11-06):

GraphData["WellsGraph","CospectralGraphNames"]

g0=GraphData["WellsGraph"];

GraphData["WellsGraph","Symmetric"]

Max[d=GraphData["WellsGraph","DistanceMatrix"]]

EdgeList[g][[1]]

xp=Position[Normal@d[[1]],4][[1,1]]

yp=Position[Normal@d[[2]],4][[1,1]]

WellsCospectralGraph1=EdgeAdd[EdgeDelete[g0,{1\[UndirectedEdge]2,24\[UndirectedEdge]30}],{1\[UndirectedEdge]24,2\[UndirectedEdge]30}]

RecognizeGraph@WellsCospectralGraph1

Tally[GraphSpectrum[WellsCospectralGraph1]]

Tally[GraphSpectrum[g0]]

Araujo-Pardo and Leemans 2022 (egr(32, 5, 5, 12))

Araujo-Pardo, G. and Leemans, D.  "Edge-Girth-Regular Graphs Arising From Biaffine Planes and Suzuki Groups."  Disc. Math. 345, 112991, 2022.

https://doi.org/10.1016/j.disc.2022.112991

https://arxiv.org/abs/2108.06636

Show[Import["https://arxiv.org/pdf/2108.06636",{"PDF","PageImages",7}],ImageSize->500]

adj=ToExpression@StringReplace[Import["https://arxiv.org/pdf/2108.06636",{"PDF","Plaintext",7}],__~~"["~~mat:("{"~~__~~"}")~~"]"~~__:>"{"<>mat<>"}"]

FromAdjacencyLists[adj]

RecognizeGraph@%

GraphData

GraphDataClassString

GraphDataClassString["BrusselsGraph"]

GraphDataString

DateString[]

file=OpenWrite["~/brussels-graph.m"]

WriteString[file,
GraphDataString[,
"StandardName"->"BrusselsGraph",
"Name"->"Brussels graph",
"Information"->"WellsGraph",
"Skip"->{"Arboricity","ConnectedInducedSubgraphPolynomial","ConstructionNumber","Cycles","CyclicEdgeConnectivity","DominatingSets","IndependentEdgeSets","IrredundancePolynomial","KTree","LinklesslyEmbeddable","MatchingGeneratingPolynomial","MinimumCyclicEdgeCuts","MinimumEdgeCuts","MinimumTotalDominatingSets","MinimumWeightFractionalEdgeColoring","MinimumVertexColorings","Minors","Perfect1Factorizations","ProjectivePlanar","Thickness","TotalDominationPolynomial","TuttePolynomial","VisibilityPolynomial"}
]]

Close[file]