Grünbaum Graphs

Author

Eric W. Weisstein
November 10, 2022

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/GruenbaumGraphs.nb.

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Figure

gruenbaum=SortBy[GraphData["Gruenbaum*"],GraphData[#,"V"]&]

GraphicsRow[Show[StyleGraphs[GraphData[#],VertexSize->{"Scaled",If[GraphData[#,"V"]==25,.025,.015]}],PlotLabel->TextCell[Style[GraphData[#,"Name"],Directive[14,Italic,Black,FontFamily->"Times"]],TextAlignment->Center]]&/@gruenbaum,ImageSize->700,Dividers->All]

Embeddings

<<MathWorld`Graphs`

"GruenbaumGraph25"

GraphData["GruenbaumGraph25","StandardName"]

Primary

GraphData["GruenbaumGraph25"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["GruenbaumGraph25","Graph","All"]//StyleGraphs

Default embedding

GraphPlot@GraphData["GruenbaumGraph25","Edges"]

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

RecognizeGraph[g]

Inexact

Select[GraphData["GruenbaumGraph25","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["GruenbaumGraph25","EdgeList"]]

Default embedding

GraphPlot@GraphData["GruenbaumGraph25","Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData["GruenbaumGraph25","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["GruenbaumGraph25","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["GruenbaumGraph25","Graph","Circular"]//StyleGraphs

Planar

GraphData["GruenbaumGraph25","Planar"]

GraphData["GruenbaumGraph25","Graph","Planar"]//StyleGraphs

Halin

GraphData["GruenbaumGraph25","Graph","Halin"]//StyleGraphs

MinimalCrossing

GraphData["GruenbaumGraph25",{"RCN","CN"}]

GraphData["GruenbaumGraph25","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData["GruenbaumGraph25"],"CrossingNumber"->26]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

LCF

GraphData["GruenbaumGraph25","LCFSignature"]

GraphData["GruenbaumGraph25","Graph","LCF"]//StyleGraphs

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

LCF compute

Length[FindHamiltonianCycle[GraphData["GruenbaumGraph25"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["GruenbaumGraph25"],All])[[All,2]]]//Timing

gs=LCFGraph@@@lcf//StyleGraphs

gs=Join[With[{g=BilateralLCFEmbedding[#]},If[MissingQ[g],#,g]]&/@LCFGraph@@@DeleteCases[lcf,{_,1}],DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]]

Length[gs]

g=BilateralLCFEmbedding[LCFGraph@@lcf[[1]]]

UnitDistance

GraphData["GruenbaumGraph25","UnitDistance"]

UnitDistanceGraphQ[GraphData["GruenbaumGraph25"],Debug->True]//Timing

GraphData["GruenbaumGraph25","Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["GruenbaumGraph25","VertexList"],GraphData["GruenbaumGraph25","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine/@GraphData["GruenbaumGraph25","Graph","All"]

Select[%,UnitDistanceGraphEmbeddingQ]

g=RotateGraph[LabelGraph@,{6,8}]

EdgeLengths[g]

UnitDistanceEmbeddingRefine/@Complement@@(GraphData["GruenbaumGraph25","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["GruenbaumGraph25"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["GruenbaumGraph25"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

FindUnitDistanceEmbedding[GraphData["GruenbaumGraph25"],MaxIterations->50000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{"GruenbaumGraph25",l=FindSubgraphUnitDistanceEmbedding[GraphData["GruenbaumGraph25"],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData["GruenbaumGraph25","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["GruenbaumGraph25","Edges"]]

3D All

GraphData["GruenbaumGraph25","Graph",{"3D","All"}]

3D UnitDistance

GraphData["GruenbaumGraph25","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["GruenbaumGraph25","Polyhedral"]

"GruenbaumGraph121"

GraphData["GruenbaumGraph121","StandardName"]

Primary

GraphData["GruenbaumGraph121"]//StyleGraphs[#,VertexSize->{"Scaled",.015}]&

All

GraphData["GruenbaumGraph121","Graph","All"]//StyleGraphs[#,VertexSize->{"Scaled",.015}]&

Select[GraphData["GruenbaumGraph121","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["GruenbaumGraph121","EdgeList"]]

Default embedding

GraphPlot@GraphData["GruenbaumGraph121","Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData["GruenbaumGraph121","Graph","Degenerate"]//StyleGraphs[#,VertexSize->{"Scaled",.015}]&

DegenerateGraphEmbeddingTypes/@GraphData["GruenbaumGraph121","Graph","Degenerate"]

Bilateral

GraphData["GruenbaumGraph121","Graph","Bilateral"]//StyleGraphs[#,VertexSize->{"Scaled",.015}]&

Circular

GraphData["GruenbaumGraph121","Graph","Circular"]//StyleGraphs[#,VertexSize->{"Scaled",.015}]&

Circular construction

Length[hc=FindHamiltonianCycle[GraphData["GruenbaumGraph121"],All][[All,All,1]]]

Length[bilat=Union@DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData["GruenbaumGraph121"],#]&/@hc]]//Timing

bilat

Length[quad=Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&]]

quad

Planar

GraphData["GruenbaumGraph121","Planar"]

GraphData["GruenbaumGraph121","Graph","Planar"]//StyleGraphs[#,VertexSize->{"Scaled",.015}]&

MinimalCrossing

GraphData["GruenbaumGraph121",{"RCN","CN"}]

GraphData["GruenbaumGraph121","Graph","MinimalCrossing"]//StyleGraphs[#,VertexSize->{"Scaled",.015}]&

MinimalCrossing construction

(cn=CrossingNumber[GraphData["GruenbaumGraph121"],TimeConstraint->1])//AbsoluteTiming

g=RotateGraph[LabelGraph@MinimalCrossingGraph[GraphData["GruenbaumGraph121"]],{120,61}]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["GruenbaumGraph121","Circulant"]

GraphData["GruenbaumGraph121","Graph","Circulant"]//StyleGraphs

Halin

GraphData["GruenbaumGraph121","Graph","Halin"]//StyleGraphs

Rigid

GraphData["GruenbaumGraph121","Rigid"]

GraphData["GruenbaumGraph121","Graph","Rigid"]//StyleGraphs[#,VertexSize->{"Scaled",.015}]&

Flexible

GraphData["GruenbaumGraph121","Flexible"]

GraphData["GruenbaumGraph121","Graph","Flexible"]//StyleGraphs[#,VertexSize->{"Scaled",.015}]&

LCF

GraphData["GruenbaumGraph121","LCFSignature"]

GraphData["GruenbaumGraph121","Graph","LCF"]//StyleGraphs

UnitDistance

GraphData["GruenbaumGraph121","UnitDistance"]

UnitDistanceGraphQ[GraphData["GruenbaumGraph121"],Debug->True]//Timing

GraphData["GruenbaumGraph121","Graph","UnitDistance"]//StyleGraphs[#,VertexSize->{"Scaled",.015}]&

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

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData["GruenbaumGraph121","Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData["GruenbaumGraph121"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{"GruenbaumGraph121",l=FindSubgraphUnitDistanceEmbedding[GraphData["GruenbaumGraph121"],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData["GruenbaumGraph121","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["GruenbaumGraph121","Edges"]]

3D All

GraphData["GruenbaumGraph121","Graph",{"3D","All"}]

3D UnitDistance

GraphData["GruenbaumGraph121","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["GruenbaumGraph121","Polyhedral"]

GraphData["GruenbaumGraph121","Graph",{"3D","Polyhedron"}]

"GruenbaumGraph124"

GraphData["GruenbaumGraph124","StandardName"]

All

GraphData["GruenbaumGraph124","Graph","All"]//StyleGraphs

Halin

GraphData["GruenbaumGraph124","Graph","Halin"]//StyleGraphs

LCF

GraphData["GruenbaumGraph124","LCFSignature"]

GraphData["GruenbaumGraph124","Graph","LCF"]//StyleGraphs

UnitDistance

GraphData["GruenbaumGraph124","UnitDistance"]

UnitDistanceGraphQ[GraphData["GruenbaumGraph124"],Debug->True]//Timing

GraphData["GruenbaumGraph124","Graph","UnitDistance"]//StyleGraphs

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

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData["GruenbaumGraph124","Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData["GruenbaumGraph124"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{"GruenbaumGraph124",l=FindSubgraphUnitDistanceEmbedding[GraphData["GruenbaumGraph124"],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData["GruenbaumGraph124","Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData["GruenbaumGraph124",{"RCN","CN"}]

GraphData["GruenbaumGraph124","Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData["GruenbaumGraph124","Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

g=RotateGraph[MinimalCrossingGraph[GraphData["GruenbaumGraph124"]],{29,49}]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData["GruenbaumGraph124","Edges"]]

3D All

GraphData["GruenbaumGraph124","Graph",{"3D","All"}]

3D UnitDistance

GraphData["GruenbaumGraph124","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["GruenbaumGraph124","Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData["GruenbaumGraph124","Polyhedron"]}]

3D Canonical

GraphData["GruenbaumGraph124","Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData["GruenbaumGraph124","Polyhedron"],Yellow,Sphere[]}]

3D Canonical construction

Graphics3D[{FaceForm[Green,Red],poly=GraphPolyhedron[GraphData["GruenbaumGraph124"],-1]}]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->1000,MaxIterations->100000,PrecisionGoal->20])//Timing

SimplePolyhedronQ[can]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

Construction

Gruenbaum25

ShowLabeledGraph[g=AddEdges[AddVertices[
Graph[List/@{{1,14},{1,15},{1,16},{2,5},{2,6},{2,13},{3,7},{3,14},{3,19},{4,8},{4,15},{4,20},{5,11},{5,19},{6,12},{6,20},{7,11},{7,16},{8,12},{8,16},{9,10},{9,14},{9,17},{10,15},{10,18},{11,12},{13,17},{13,18},{17,19},{18,20}},List/@{{-1.628`,0.`},{1.192`,0.`},{-0.503`,-1.548`},{-0.503`,1.548`},{1.317`,-0.957`},{1.317`,0.957`},{-0.762`,-2.345`},{-0.762`,2.345`},{-0.416`,-0.302`},{-0.416`,0.302`},{1.995`,-1.449`},{1.995`,1.449`},{0.514`,0.`},{-0.964`,-0.701`},{-0.964`,0.701`},{-2.466`,0.`},{0.159`,-0.489`},{0.159`,0.489`},{0.368`,-1.134`},{0.368`,1.134`}}],3.2Through[{Cos,Sin}[2π (#+1/2.)/5]]&/@Range[0,4]],
{
{12,21},{8,22},{16,23},{7,24},{11,25},
{21,15},{21,13},{21,3},
{25,20},{25,17},{25,1},
{24,2},{24,4},{24,9},
{23,19},{23,6},{23,10},
{22,14},{22,18},{22,5}
}
]
]

FastIsomorphicQ[g,Graph["GruenbaumGraph"]]

Gruenbaum121

ShowGraph[g2=DeleteVertices[g,Range[3]],VertexStyle->Disk[.02]]

FindHamiltonianCycle[g2]//Timing

HamiltonianCycles[g2]//Timing

Print[GraphDataString[g2,
"Name"->"Grünbaum graph 121",
"StandardName"->"GruenbaumGraph121",
"AlternateStandardNames"->{"GrunbaumGraph121"},
TimeConstraint->60,
Debug->True]
]

Gruenbaum124

g=Graph[{{{1,2}},{{1,3}},{{1,4}},{{2,3}},{{2,60}},{{3,117}},{{4,5}},{{4,6}},{{5,6}},{{5,7}},{{6,17}},{{7,8}},{{7,9}},{{8,9}},{{8,58}},{{9,11}},{{10,11}},{{10,12}},{{10,26}},{{11,12}},{{12,13}},{{13,14}},{{13,15}},{{14,15}},{{14,20}},{{15,16}},{{16,17}},{{16,18}},{{17,18}},{{18,114}},{{19,20}},{{19,21}},{{19,39}},{{20,21}},{{21,22}},{{22,23}},{{22,24}},{{23,24}},{{23,35}},{{24,25}},{{25,26}},{{25,27}},{{26,27}},{{27,28}},{{28,29}},{{28,30}},{{29,30}},{{29,49}},{{30,31}},{{31,32}},{{31,33}},{{32,33}},{{32,34}},{{33,44}},{{34,35}},{{34,36}},{{35,36}},{{36,41}},{{37,38}},{{37,39}},{{37,40}},{{38,39}},{{38,105}},{{40,41}},{{40,42}},{{41,42}},{{42,100}},{{43,44}},{{43,84}},{{43,100}},{{44,45}},{{45,46}},{{45,47}},{{46,47}},{{46,48}},{{47,75}},{{48,49}},{{48,50}},{{49,50}},{{50,51}},{{51,52}},{{51,53}},{{52,53}},{{52,72}},{{53,54}},{{54,55}},{{54,56}},{{55,56}},{{55,68}},{{56,57}},{{57,58}},{{57,59}},{{58,59}},{{59,61}},{{60,61}},{{60,62}},{{61,62}},{{62,63}},{{63,64}},{{63,65}},{{64,65}},{{64,123}},{{65,66}},{{66,67}},{{66,68}},{{67,68}},{{67,69}},{{69,70}},{{69,71}},{{70,71}},{{70,78}},{{71,74}},{{72,73}},{{72,74}},{{73,74}},{{73,77}},{{75,76}},{{75,77}},{{76,77}},{{76,82}},{{78,79}},{{78,80}},{{79,80}},{{79,89}},{{80,81}},{{81,82}},{{81,83}},{{82,83}},{{83,84}},{{84,85}},{{85,86}},{{85,87}},{{86,87}},{{86,99}},{{87,88}},{{88,89}},{{88,90}},{{89,90}},{{90,91}},{{91,92}},{{91,93}},{{92,93}},{{92,120}},{{93,94}},{{94,95}},{{94,96}},{{95,96}},{{95,108}},{{96,97}},{{97,98}},{{97,99}},{{98,99}},{{98,103}},{{100,101}},{{101,102}},{{101,103}},{{102,103}},{{102,104}},{{104,105}},{{104,106}},{{105,106}},{{106,107}},{{107,108}},{{107,109}},{{108,109}},{{109,110}},{{110,111}},{{110,112}},{{111,112}},{{111,113}},{{112,119}},{{113,114}},{{113,115}},{{114,115}},{{115,116}},{{116,117}},{{116,118}},{{117,118}},{{118,124}},{{119,120}},{{119,121}},{{120,121}},{{121,122}},{{122,123}},{{122,124}},{{123,124}}},{{{0.547808147663403,0.9904442309624726}},{{1.1422643797193497,-0.011804633090502573}},{{-0.02780828807293329,-0.01158632922742982}},{{0.5516028319405757,0.760063834726091}},{{0.5516028319405757,0.7300638347260909}},{{0.5282694986072423,0.7233971680594242}},{{0.6432694986072424,0.6850638347260909}},{{0.669936165273909,0.6717305013927576}},{{0.6582694986072423,0.6567305013927576}},{{0.6366028319405757,0.5950638347260909}},{{0.639936165273909,0.6267305013927575}},{{0.6132694986072423,0.6283971680594242}},{{0.4882694986072423,0.6283971680594242}},{{0.4582694986072423,0.6100638347260909}},{{0.46493616527390896,0.630063834726091}},{{0.4432694986072423,0.6567305013927576}},{{0.45493616527390895,0.6883971680594243}},{{0.424936165273909,0.6700638347260909}},{{0.45160283194057566,0.5417305013927576}},{{0.45493616527390895,0.5633971680594242}},{{0.4782694986072423,0.5533971680594243}},{{0.5166028319405757,0.5433971680594243}},{{0.5532694986072423,0.5067305013927575}},{{0.549936165273909,0.5317305013927576}},{{0.6166028319405756,0.5583971680594243}},{{0.6382694986072424,0.5617305013927576}},{{0.6466028319405757,0.5283971680594243}},{{0.6632694986072423,0.4900638347260909}},{{0.6782694986072423,0.4600638347260909}},{{0.659936165273909,0.4550638347260909}},{{0.639936165273909,0.45173050139275756}},{{0.6182694986072423,0.44339716805942425}},{{0.6199361652739089,0.42339716805942423}},{{0.5466028319405757,0.4600638347260909}},{{0.5482694986072423,0.48339716805942423}},{{0.5232694986072423,0.45173050139275756}},{{0.4482694986072423,0.45339716805942426}},{{0.41826949860724233,0.4633971680594242}},{{0.429936165273909,0.48839716805942424}},{{0.46493616527390896,0.44339716805942425}},{{0.4816028319405756,0.4350638347260909}},{{0.4766028319405756,0.4100638347260909}},{{0.549936165273909,0.3317305013927576}},{{0.6299361652739089,0.3617305013927576}},{{0.659936165273909,0.33506383472609097}},{{0.6982694986072423,0.34173050139275757}},{{0.674936165273909,0.32339716805942426}},{{0.7116028319405756,0.3567305013927576}},{{0.7134346703957765,0.38306682448235574}},{{0.7399361652739089,0.3517305013927576}},{{0.7766028319405757,0.3033971680594243}},{{0.7766028319405757,0.26839716805942426}},{{0.794936165273909,0.2817305013927576}},{{0.8182694986072423,0.2667305013927576}},{{0.8232694986072423,0.2267305013927576}},{{0.8416028319405756,0.25506383472609095}},{{0.874936165273909,0.25339716805942425}},{{0.9016028319405758,0.2567305013927576}},{{0.8949361652739091,0.23339716805942426}},{{0.8949361652739091,0.11339716805942429}},{{0.879936165273909,0.14839716805942427}},{{0.874936165273909,0.12339716805942426}},{{0.7932694986072423,0.06006383472609095}},{{0.769936165273909,0.05339716805942428}},{{0.764936165273909,0.07173050139275762}},{{0.7449361652739089,0.09339716805942429}},{{0.724936165273909,0.10839716805942429}},{{0.764936165273909,0.11673050139275762}},{{0.6932694986072423,0.12173050139275762}},{{0.6716028319405757,0.12339716805942426}},{{0.699936165273909,0.14006383472609094}},{{0.7482694986072423,0.23339716805942426}},{{0.704936165273909,0.226730501392757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