Crossed Graph

Author

Eric W. Weisstein
June 26, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Table

polys=Union[Select[
Union@@(PolyhedronData/@{"Platonic","Archimedean","ArchimedeanDual"}),PolyhedronData[#,"Faces","Tally"][[All,1]]=!={3}&
],{"ElongatedSquareGyrobicupola"}]

TextGrid[
{
#,
Show[PolyhedronData[#],Boxed->False],
g=Annotate[CrossedGraph[PolyhedronData[#,"Polyhedron"]],ImageSize->Small],
RecognizeGraph[g]
}&/@polys,Dividers->All]

Embeddings

<<MathWorld`Graphs`

"CrossedCubicalGraph" == "SixteenCellGraph"

Names

GraphData["CrossedCubicalGraph","StandardName"]

GraphData["CrossedCubicalGraph","StandardNames"]

"CrossedCuboctahedralGraph" == "OctahedralLineGraph"

Names

GraphData["CrossedCuboctahedralGraph","StandardName"]

GraphData["CrossedCuboctahedralGraph","StandardNames"]

"CrossedDodecahedralGraph"

Names

GraphData["CrossedDodecahedralGraph","StandardName"]

GraphData["CrossedDodecahedralGraph","StandardNames"]

Construction

Brandenburg, F.~J.  ``Straight-Line Drawings of 1-Planar Graphs.''  3 Sep 2021.  https://arxiv.org/abs/2109.01692

Import["/Users/eww/Downloads/arXiv-2109.01692v1/Xdodecaeder.pdf"][[1]]

ee=Union[Sort/@Flatten[Subsets[#,{2}]&/@GraphData["DodecahedralGraph","Faces"],1]]

g=Graph[Range[20],UndirectedEdge@@@ee,VertexCoordinates->GraphData["DodecahedralGraph","VertexCoordinates"],VertexLabels->Automatic]

RecognizeGraph@g

gs=Graph[Range[20],UndirectedEdge@@@ee,VertexCoordinates->#,VertexLabels->Automatic]&/@GraphData["DodecahedralGraph","Embeddings"]

Select[gs,(!DegenerateGraphEmbeddingQ[#])&]

gs=UnlabelGraph/@{,,,,,,,}

Head/@gs

Primary

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

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["CrossedDodecahedralGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Configuration

GraphData["CrossedDodecahedralGraph","Configuration"]

GraphData["CrossedDodecahedralGraph","Graph","Configuration"]//StyleGraphs

Perspective

GraphData["CrossedDodecahedralGraph","Graph","Perspective"]//StyleGraphs

Projection construction

Manipulate[ProjectionGraph[GraphData["CrossedDodecahedralGraph","Graph",{"3D"}],{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1}]

ProjectionGraph[GraphData["CrossedDodecahedralGraph","Graph","3D"],{0,.01,.01}]

ProjectionGraph[GraphData["CrossedDodecahedralGraph","Graph","3D"],{0,0,1}]

DegenerateGraphEmbeddingTypes@

DegenerateGraphEmbeddingTypes[%]

ProjectionGraph[GraphData["CrossedDodecahedralGraph","Graph","3D"],{0,1,0}]

DegenerateGraphEmbeddingTypes[%]

ProjectionGraph[GraphData["CrossedDodecahedralGraph","Graph","3D"],{1,0.01,0}]

ProjectionGraph[GraphData["CrossedDodecahedralGraph","Graph","3D"],{1,1,0}]

ProjectionGraph[GraphData["CrossedDodecahedralGraph","Graph","3D"],{1,0,1,-Pi/2}]

ProjectionGraph[GraphData["CrossedDodecahedralGraph","Graph","3D"],{1,1,1}]

gs={,,,};

HoneycombToroidal construction

GraphData["CrossedDodecahedralGraph","NotationRules","HoneycombToroidal"]

GeneralizedPetersen

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

IGraph

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

Bilateral

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

Circular

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

Planar

GraphData["CrossedDodecahedralGraph","Planar"]

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

MinimalLocalCrossing

Brandenburg, F.~J.  ``Straight-Line Drawings of 1-Planar Graphs.''  3 Sep 2021.  https://arxiv.org/abs/2109.01692

GraphData["CrossedDodecahedralGraph","LocalCrossingNumber"]

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

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

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossedDodecahedralGraph"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossedDodecahedralGraph","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["CrossedDodecahedralGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union//Timing

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossedDodecahedralGraph","Circulant"]

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

Complement graph

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

GraphData[compl,"Graph","All"]

gs=GraphComplement/@%

Line graph

GraphData["CrossedDodecahedralGraph","Line"]

LineGraphQ[GraphData["CrossedDodecahedralGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["CrossedDodecahedralGraph"]]

Halin

GraphData["CrossedDodecahedralGraph","Halin"]

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

Rigid

GraphData["CrossedDodecahedralGraph","Rigid"]

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

Flexible

GraphData["CrossedDodecahedralGraph","Flexible"]

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

LCF

GraphData["CrossedDodecahedralGraph","LCF"]

GraphData["CrossedDodecahedralGraph","HamiltonianCycleCount"]

GraphData["CrossedDodecahedralGraph","LCFSignature"]

GraphData["CrossedDodecahedralGraph","LCFNotations"]

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

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

LCF compute random

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

LCF compute 10^5

Tally[(lcf=LCF[GraphData["CrossedDodecahedralGraph"],10^5])[[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

LCF compute 10^6

Tally[(lcf=LCF[GraphData[{"NoncayleyTransitive",{20,41}}],10^6])[[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

LCF compute 10^7

Tally[(lcf=LCF[GraphData[{"NoncayleyTransitive",{20,41}}],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

LCF compute 10^8

Tally[(lcf=LCF[GraphData[{"NoncayleyTransitive",{20,41}}],10^8])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1|2}]

LCFGraph@@@DeleteCases[lcf,{_,1|2}]//StyleGraphs

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

Length[gs]

gs

LCF compute all

Divisors[GraphData["CrossedDodecahedralGraph","V"]]

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

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

MaxMemoryUsed[]/10.^9 "GB"

Tally[(lcf=ReadProperty["LCFNotations",GraphNameToFileName["CrossedDodecahedralGraph",".m.gz"]])[[All,2]]]//Timing

DeleteCases[lcf,{_,1}]

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

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

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

i=0;
gs={};
t0=AbsoluteTime[];
count=Length[order2];
Monitor[
Do[
If[!MissingQ[g=BilateralLCFGraph[LCFGraph@@order2[[i]]]],AppendTo[gs,g]],
{i,count}
],
Column[{
ProgressIndicator[i,{1,count}],
Row[{i,"/",count}],
Row[{"estimated time remaining: ",Round[(AbsoluteTime[]-t0) (count/i-1)]," s"}],
Row[{Length[gs]," bilateral graphs"}]
}]
];//Timing

UnitDistance

GraphData["CrossedDodecahedralGraph","UnitDistance"]

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossedDodecahedralGraph","VertexList"],GraphData["CrossedDodecahedralGraph","EdgeList"]],MaxIterations->500]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["CrossedDodecahedralGraph","Graph",#]&/@{"All","UnitDistance"})

LabelGraph@

DegenerateGraphEmbeddingTypes@

gs={RotateGraph[,{5,12}],RotateGraph[,{2,6}],RotateGraph[,{9,10}]}

UnitDistance by FindUnitDistanceEmbedding

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

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

Matchstick

GraphData["CrossedDodecahedralGraph","Matchstick"]

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

3D default

Graph3D[GraphData["CrossedDodecahedralGraph","Edges"]]

3D construction

v=PolyhedronData["Dodecahedron","VertexCoordinates"];

ee=Union[Sort/@Flatten[Subsets[#,{2}]&/@PolyhedronData["Dodecahedron","Faces","List"],1]]

g=Graph[Range[20],UndirectedEdge@@@ee,VertexCoordinates->v]

RecognizeGraph@g

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["CrossedDodecahedralGraph","Polyhedral"]

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

"CrossedElongatedSquareGyrobicupolaGraph" == "FabriciMadarasGraph24"

Names

GraphData["CrossedElongatedSquareGyrobicupolaGraph","StandardName"]

GraphData["CrossedElongatedSquareGyrobicupolaGraph","StandardNames"]

Construction

Fabrici, I. and Madaras, T.  ``The Structure of 1-Planar Graphs.''  {\it Disc.\ Math.} {\bf 307}, 854--865, 2007.  https://doi.org/10.1016/j.disc.2005.11.056

v=Join[Union[
v1=Flatten[Table[{i,j},{i,-4,4,2},{j,{-1,1}}],1],
Reverse/@v1
],CirclePoints[8,4]];
g=Graph[Range@Length@v,UndirectedEdge@@@{e[5,12,13,14,15,16,9,8,7,6,5,13,17,20,16,8,4,1,5,3,4,9],e[12,19,20,18,11,4,2,1,3,10,17,19,18,15,9,23,2],e[3,7,11,15],e[17,14,11,8],e[4,7,10,14,18],e[2,3,6,12,17,18,16,22,20],e[1,6,13,19],e[1,24,5],e[12,21,19],e[6,10,13],{10,11},{7,14},{2,9},{24,21},{12,24},{5,21},{20,21},{2,24},{21,22},{24,23},{24,22},{21,23},{22,23},{9,22},{16,23},{1,23},{19,22},e[2,8,15,20]},VertexCoordinates->v,VertexLabels->Automatic]

VertexDegree[g]//Union

RecognizeGraph@g

PlanarGraphQ@g

Primary

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

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["CrossedElongatedSquareGyrobicupolaGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Configuration

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Configuration"]

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph","Configuration"]//StyleGraphs

Perspective

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph","Perspective"]//StyleGraphs

Projection construction

Manipulate[ProjectionGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph",{"3D"}],{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1}]

ProjectionGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph","3D"],{0,.01,.01}]

ProjectionGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph","3D"],{0,0,1}]

ProjectionGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph","3D"],{0,1,0}]

ProjectionGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph","3D"],{1,0.01,0}]

ProjectionGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph","3D"],{1,1,0}]

ProjectionGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph","3D"],{1,0,1,-Pi/2}]

ProjectionGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph","3D"],{1,1,1}]

gs={, ,};

HoneycombToroidal construction

GraphData["CrossedElongatedSquareGyrobicupolaGraph","NotationRules","HoneycombToroidal"]

GeneralizedPetersen

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

IGraph

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

Bilateral

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

Circular

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

Planar

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Planar"]

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

MinimalLocalCrossing

GraphData["CrossedElongatedSquareGyrobicupolaGraph","LocalCrossingNumber"]

Fabrici, I. and Madaras, T.  ``The Structure of 1-Planar Graphs.''  {\it Disc.\ Math.} {\bf 307}, 854--865, 2007.  https://doi.org/10.1016/j.disc.2005.11.056

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

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

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph"],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData["CrossedElongatedSquareGyrobicupolaGraph","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,500}]//Union//Timing

GraphPlot@

v[[19,1]]

v[[14,2]]//FullSimplify

1/3+1/12

v=Join[CirclePoints[3],CirclePoints[1/2-1/12,6],CirclePoints[1/3-1/24,7],CirclePoints[1/10,4],{{-1/5,0},{1/5,0},{-(1/(10 (Sqrt[2]))),(1/3) Sin[((3 π)/14)]},{(1/(10 (Sqrt[2]))),(1/3) Sin[((3 π)/14)]}}];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,1,5,6,2,7,8,3,9,4,1,6,7,3,4,16,10,9,1,11,17,20,15,3],e[9,8,14,7,13,6,12,5,4,10,11,5],e[9,16,15,8],e[2,13,14,23,24,12,13],e[9,15,21,19,18,22,11,4],e[16,20,19,23,7],e[10,17,18,24,6],e[21,23,13,24,22],{21,14},{12,22},{11,12},{15,14},e[8,2,5],{14,19},{18,12},{20,18},{17,19},{19,24},{18,23},{16,17},{20,10},{5,22},{8,21},{10,22},{16,21},{17,22},{20,21}},VertexCoordinates->v,VertexLabels->Automatic,VertexSize->{"Scaled",.01}]

RectilinearCrossingCount[g]

LocalCrossingCount[g]

Circulant

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Circulant"]

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

Complement graph

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

Line graph

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Line"]

LineGraphQ[GraphData["CrossedElongatedSquareGyrobicupolaGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["CrossedElongatedSquareGyrobicupolaGraph"]]

RecognizeGraph[%]

Halin

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Halin"]

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

Rigid

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Rigid"]

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

Flexible

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Flexible"]

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

LCF

GraphData["CrossedElongatedSquareGyrobicupolaGraph","LCF"]

GraphData["CrossedElongatedSquareGyrobicupolaGraph","HamiltonianCycleCount"]

GraphData["CrossedElongatedSquareGyrobicupolaGraph","LCFSignature"]

GraphData["CrossedElongatedSquareGyrobicupolaGraph","LCFNotations"]

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

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

LCF compute random

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

LCF compute 10^5

Tally[(lcf=LCF[GraphData["CrossedElongatedSquareGyrobicupolaGraph"],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

LCF compute 10^6

Tally[(lcf=LCF[GraphData["CrossedElongatedSquareGyrobicupolaGraph"],10^6])[[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

LCF compute 10^7

Tally[(lcf=LCF[GraphData["CrossedElongatedSquareGyrobicupolaGraph"],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

LCF compute all

Divisors[GraphData["CrossedElongatedSquareGyrobicupolaGraph","V"]]

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

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

MaxMemoryUsed[]/10.^9 "GB"

Tally[(lcf=ReadProperty["LCFNotations",GraphNameToFileName["CrossedElongatedSquareGyrobicupolaGraph",".m.gz"]])[[All,2]]]//Timing

DeleteCases[lcf,{_,1}]

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

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

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

i=0;
gs={};
t0=AbsoluteTime[];
count=Length[order2];
Monitor[
Do[
If[!MissingQ[g=BilateralLCFGraph[LCFGraph@@order2[[i]]]],AppendTo[gs,g]],
{i,count}
],
Column[{
ProgressIndicator[i,{1,count}],
Row[{i,"/",count}],
Row[{"estimated time remaining: ",Round[(AbsoluteTime[]-t0) (count/i-1)]," s"}],
Row[{Length[gs]," bilateral graphs"}]
}]
];//Timing

UnitDistance

GraphData["CrossedElongatedSquareGyrobicupolaGraph","UnitDistance"]

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossedElongatedSquareGyrobicupolaGraph","VertexList"],GraphData["CrossedElongatedSquareGyrobicupolaGraph","EdgeList"]],MaxIterations->500]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["CrossedElongatedSquareGyrobicupolaGraph","Graph",#]&/@{"All","UnitDistance"})

UnitDistance by FindUnitDistanceEmbedding

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

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

Matchstick

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Matchstick"]

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

3D default

Graph3D[GraphData["CrossedElongatedSquareGyrobicupolaGraph","Edges"]]

3D construction

g=CrossedGraph[PolyhedronData["ElongatedSquareGyrobicupola","Polyhedron"]]

RecognizeGraph@g

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["CrossedElongatedSquareGyrobicupolaGraph","Polyhedral"]

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

"CrossedIcosidodecahedralGraph" == "IcosahedralLineGraph"

Names

GraphData["CrossedIcosidodecahedralGraph","StandardName"]

GraphData["CrossedIcosidodecahedralGraph","StandardNames"]

"CrossedTruncatedTetrahedralGraph" == {"Circulant", {12, {1, 2, 4, 5, 6}}}

Names

GraphData[{"Circulant",{12,{1,2,4,5,6}}},"StandardName"]

GraphData["CrossedTruncatedTetrahedralGraph","StandardNames"]

GraphData

GraphDataClassString

GraphDataClassString["CrossedDodecahedralGraph"]