Dodecicosahedral Graph

Author

Eric W. Weisstein
July 21, 2026

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

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

©2026 Wolfram Research, Inc. except for portions noted otherwise

Graphic

GraphicsGrid[Partition[StyleGraphs@GraphData["DodecicosahedralGraph","Graph","All"],UpTo[6]],Dividers->All]

Embeddings

<<MathWorld`Graphs`

Names

GraphData["DodecicosahedralGraph","StandardName"]

GraphData["DodecicosahedralGraph","StandardNames"]

HouseOfGraphs

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

HouseOfGraphsLookup[hogid]

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

Primary

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

All

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

Inexact

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

Degenerate

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

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

Embedding types

GraphDataEmbeddings[GraphData["DodecicosahedralGraph","EdgeList"]]

Default embedding

g=Graph[GraphData["DodecicosahedralGraph","Edges"],GraphLayout->"SpringEmbedding"]

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

RecognizeGraph[g]

Perspective

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

Perspective construction

GraphData["DodecicosahedralGraph","Graph","3D"]

PolyhedronGraph[GraphData["DodecicosahedralGraph","Polyhedron"]]

Manipulate[PolyhedronProjectionGraph[GraphData["DodecicosahedralGraph","Polyhedron"],{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1}]

PolyhedronProjectionGraph[GraphData["DodecicosahedralGraph","Polyhedron"],{0,.01,.01}]

PolyhedronProjectionGraph[GraphData["DodecicosahedralGraph","Polyhedron"],{0,0,1}]

PolyhedronProjectionGraph[GraphData["DodecicosahedralGraph","Polyhedron"],{0,1,0}]

PolyhedronProjectionGraph[GraphData["DodecicosahedralGraph","Polyhedron"],{1,0.01,0}]

PolyhedronProjectionGraph[GraphData["DodecicosahedralGraph","Polyhedron"],{1,1,0}]

PolyhedronProjectionGraph[GraphData["DodecicosahedralGraph","Polyhedron"],{1,0,1,-Pi/2}]

PolyhedronProjectionGraph[GraphData["DodecicosahedralGraph","Polyhedron"],{1,1,1}]

Bipartite

GraphData["DodecicosahedralGraph","Bipartite"]

GraphData["DodecicosahedralGraph","Graph","Bipartite"]//StyleGraphs

Bilateral

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

Circular

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

Planar

GraphData["DodecicosahedralGraph","Planar"]

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

MinimalCrossingNumber crossings.uos.de [failed]

CrossingNumber[GraphData["DodecicosahedralGraph"],Method->"WebCompute"]//AbsoluteTiming

DateString[]

MinimalCrossingNumber: "EmbeddingScheme" -> 1

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 2

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 3

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

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

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

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

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

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

MinimalCrossing

GraphData["DodecicosahedralGraph","CN"]

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

GraphCrossingCount/@gs

GraphCrossingDiagram/@gs

MinimalRectilinearCrossing

GraphData["DodecicosahedralGraph","RCN"]

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

MinimalCrossing construction Tutte

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

(g=MinimalCrossingGraph[GraphData["DodecicosahedralGraph"],"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["DodecicosahedralGraph","V"]])]
}&/@TutteGraphs;

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

Cases[TutteGraphsWithOuterCrossingPointCounts,{g_,1}:>g]

LocalCrossingNumberWithEmbeddings=
{1+Max[Length/@DeleteCases[
Drop[AdjacencyList[#],GraphData["DodecicosahedralGraph","V"]],
_Integer?(LessEqualThan[GraphData["DodecicosahedralGraph","V"]]),
{2}
]],#}&/@TutteGraphs

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

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

Select[SymmetricEmbeddingsWithOrientations[[All,1]],Count[OuterFace[#],_?(GreaterThan[GraphData["DodecicosahedralGraph","V"]])]==0&]

MinimalCrossing construction Random 1k [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["DodecicosahedralGraph"],
1000,
"CrossingNumber"->cn,
EdgeShapeFunction->"BezierEdge"
]]//AbsoluteTiming

No crossing points on the boundary:

Length[diagrams0=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==0&],"Graph",{}]]//Timing

Take[diagrams0,UpTo[10]]

One crossing point on the boundary:

Length[diagrams1=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==1&],"Graph",{}]]//Timing

Take[diagrams1,UpTo[10]]

graphs=#["Graph"]&/@diagrams;

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[#]
}&/@graphs;//AbsoluteTiming

MaxMemoryUsed[]/10^9.GB

Symmetric embeddings:

Length[SymmetricMinimalEmbeddingData=DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]
]//AbsoluteTiming

SymmetricMinimalEmbeddingData

SymmetricMinimalEmbeddingGraphs=SymmetricMinimalEmbeddingData[[All,1]]

GraphCrossingCount/@SymmetricMinimalEmbeddingGraphs

GraphCrossingDiagram/@SymmetricMinimalEmbeddingGraphs

MinimalCrossing construction Random 5k [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["DodecicosahedralGraph"],
5000,
"CrossingNumber"->cn,
EdgeShapeFunction->"BezierEdge"
]]//AbsoluteTiming

No crossing points on the boundary:

Length[diagrams0=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==0&],"Graph",{}]]//Timing

Take[diagrams0,UpTo[10]]

One crossing point on the boundary:

Length[diagrams1=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==1&],"Graph",{}]]//Timing

Take[diagrams1,UpTo[10]]

graphs=#["Graph"]&/@diagrams;

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[#]
}&/@graphs;//AbsoluteTiming

MaxMemoryUsed[]/10^9.GB

Symmetric embeddings:

Length[SymmetricMinimalEmbeddingData=DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]
]//AbsoluteTiming

SymmetricMinimalEmbeddingData

SymmetricMinimalEmbeddingGraphs=SymmetricMinimalEmbeddingData[[All,1]]

GraphCrossingCount/@SymmetricMinimalEmbeddingGraphs

GraphCrossingDiagram/@SymmetricMinimalEmbeddingGraphs

MinimalCrossing construction Random 20k [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["DodecicosahedralGraph"],
20*^3,
"CrossingNumber"->cn,
EdgeShapeFunction->"BezierEdge"
]]//AbsoluteTiming

No crossing points on the boundary:

Length[diagrams0=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==0&],"Graph",{}]]//Timing

Take[diagrams0,UpTo[10]]

One crossing point on the boundary:

Length[diagrams1=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==1&],"Graph",{}]]//Timing

Take[diagrams1,UpTo[10]]

graphs=#["Graph"]&/@diagrams;

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[#]
}&/@graphs;//AbsoluteTiming

MaxMemoryUsed[]/10^9.GB

Symmetric embeddings:

Length[SymmetricMinimalEmbeddingData=DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]
]//AbsoluteTiming

SymmetricMinimalEmbeddingData

SymmetricMinimalEmbeddingGraphs=SymmetricMinimalEmbeddingData[[All,1]]

GraphCrossingCount/@SymmetricMinimalEmbeddingGraphs

GraphCrossingDiagram/@SymmetricMinimalEmbeddingGraphs

MinimalCrossing construction Random 1k [old]

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

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

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

Small numbers of crossing points on the boundary:

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

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,1}:>g]

Symmetric embeddings:

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

MaxMemoryUsed[]/10^9.GB

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

MinimalCrossing construction Random 20k [old]

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

(RandomGraphs=Table[MinimalCrossingGraph[GraphData["DodecicosahedralGraph"],"CrossingNumber"->cn,RandomSeed->n],{n,20000}]//Union);//Timing

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

Small numbers of crossing points on the boundary:

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

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,1}:>g]

Symmetric embeddings:

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

MaxMemoryUsed[]/10^9.GB

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

MinimalLocalCrossing

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

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

MinimalRectilinearLocalCrossing

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

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

MinimaRectilinearlLocalCrossing construction 1000

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

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

(gs=Select[Table[MinimalCrossingGraph[GraphData["DodecicosahedralGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,1000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

MinimaRectilinearlLocalCrossing construction 20000

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

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

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

Circulant

GraphData["DodecicosahedralGraph","Circulant"]

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

Line graph

GraphData["DodecicosahedralGraph","Line"]

LineGraphQ[GraphData["DodecicosahedralGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["DodecicosahedralGraph"]]

Halin

GraphData["DodecicosahedralGraph","Halin"]

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

Rigid

GraphData["DodecicosahedralGraph","Rigid"]

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

Flexible

GraphData["DodecicosahedralGraph","Flexible"]

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

Voltage

GraphData["DodecicosahedralGraph","Graph","Voltage"]//StyleGraphs

Voltage construction

gs=VoltageGraph[GraphData["DodecicosahedralGraph"],"MaxResults"->20,Debug->True]

Length[gs]

RecognizeGraph/@gs

VoltageGraphNotation/@gs//Column

EmbeddingClasses/@gs//Column

GraphSymmetryGroup/@gs//Column

LCF

GraphData["DodecicosahedralGraph","LCF"]

GraphData["DodecicosahedralGraph","HamiltonianCycleCount"]

GraphData["DodecicosahedralGraph","LCFSignature"]

GraphData["DodecicosahedralGraph","LCFNotations"]

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

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

LCF compute FindBilateralHamiltonianCycle

BilateralHamiltonianCycleQ[GraphData["DodecicosahedralGraph"]]//Timing

(BilateralHamiltonianCycles=FindBilateralHamiltonianCycle[GraphData["DodecicosahedralGraph"]])//Timing

LCF compute FindHighlySymmetricLCFNotation

lcf=FindHighlySymmetricLCFNotation[GraphData["DodecicosahedralGraph"],TimeConstraint->600,Debug->True];//AbsoluteTiming

lcf[[All,-1]]//Tally

gs=NiceLCFGraphs[lcf,5,2]

nicelcf=LCFNotation/@gs

nicelcftally=nicelcf[[All,-1]]//Tally

nicelcftally[[All,-1]]//Total

l={
{"DodecicosahedralGraph",{0,Missing["NotAvailable",Cases[nicelcf,{_,_}]]}}
}

PropertyReplace[l,"LCFNotations"]

RenameGraphFiles[]

l2={
{"DodecicosahedralGraph",{0,Missing["NotAvailable",{{10,4},{6,6},{5,4},{3,36},{2,19533}}]}}
}

PropertyReplace[l2,"LCFSignature"]

RenameGraphFiles[]

bilatgs=Select[gs,BilateralGraphEmbeddingQ]

bilatlcf=LCFNotation/@bilatgs;

bilatlcf[[All,-1]]//Tally

LCF compute random

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

LCF compute 10^5

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

DeleteCases[lcf,{_,1}]

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

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,XX}]]}])//Timing

Length[gs]

LCF compute 10^6

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

DeleteCases[lcf,{_,1}]

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

(gs=Flatten[{
(* all LCF's with order >2, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1|2}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,2}]]}])//Timing

Length[gs]

LCF compute 10^7

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

DeleteCases[lcf,{_,1}]

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

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

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1|2}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,2}]]}])//Timing

Length[gs]

LCF compute 5*^7 [227 GB]

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

MaxMemoryUsed[]/10.^9"GB"

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

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

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1|2}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,XXX}]]}])//Timing

Length[gs]

LCF compute all

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

MaxMemoryUsed[]/10.^9 "GB"

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

MaxMemoryUsed[]/10.^9 "GB"

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

DeleteCases[lcf,{_,1}]

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

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

UnitDistance

GraphData["DodecicosahedralGraph","UnitDistance"]

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

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

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

UnitDistance by refinement of default embeddings

Graph[edge list]

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

UnitDistanceGraphEmbeddingQ[g]

Graph[vertex list, edge list]

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

UnitDistanceGraphEmbeddingQ[g]

g=RotateGraph[,.45]

Max[Abs[EdgeLengths[g]-1]]

UnitDistance by refinement of existing embeddings

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

Select[gs,UnitDistanceGraphEmbeddingQ]

GraphsWithSymmetryGroups=DeleteCases[{BilateralGraph[CenterGraph@#,"ReturnNonbilateralGraph"->True],GraphSymmetryGroup[#]}&/@gs,{_,TransformationGroup[{}]}]

NondegenrateSymmetricGraphs=Select[GraphsWithSymmetryGroups[[All,1]],!DegenerateGraphEmbeddingQ[#]&]

Max[Abs[EdgeLengths[#]-1]]&/@NondegenrateSymmetricGraphs

UnitDistance by FindUnitDistanceEmbedding  (random embeddings)

(g=FindUnitDistanceEmbedding[GraphData["DodecicosahedralGraph"],MaxIterations->5000,"ReturnWithMessage"->True])//Timing

UnitDistanceGraphEmbeddingQ[g]

(gs=FindUnitDistanceEmbedding[GraphData["DodecicosahedralGraph"],50,MaxIterations->5000,"ReturnWithMessage"->True])//Timing

Select[gs,UnitDistanceGraphEmbeddingQ]

Select[gs,GraphSymmetryGroup[#]=!=TransformationGroup[{}]&]

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

{$Version,DateString[]}

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

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

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

Matchstick

GraphData["DodecicosahedralGraph","Matchstick"]

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

3D default

Graph3D[GraphData["DodecicosahedralGraph","Edges"]]

3D All

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

3D UnitDistance construction from uniform polyhedron

emb=GraphData["DodecicosahedralGraph","PolyhedralEmbeddings"]

PolyhedronGraph[PolyhedronData[#,"Polyhedron"]]&/@emb

3D UnitDistance

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

3D Polyhedron

GraphData["DodecicosahedralGraph","Polyhedral"]

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

Properties

PolyhedralEmbeddings

GraphData["DodecicosahedralGraph","PolyhedralEmbeddings"]

GraphData

GraphDataClassString

GraphDataClassString["DodecicosahedralGraph"]

GraphDataString

file=OpenAppend["~/polygraphs.m"]

g=PolyhedronGraph[PolyhedronData["GreatDodecicosahedron","Polyhedron"]]

lowercasedecamel[s_]:=StringTrim[StringReplace[s,RegularExpression["([A-Z])"]:>" "<>ToLowerCase["$1"]]]

PolyhedronNameToGraphName[s_]:=StringReplace[s,"dron"->"dralGraph"]

With[
{
stdname="DodecicosahedralGraph",altnames={"GreatDitrigonalDodecicosidodecahedron","GreatDodecicosahedron","GreatIcosicosidodecahedron","SmallDitrigonalDodecicosidodecahedron","SmallDodecicosahedron","SmallIcosicosidodecahedron"}
},
WriteString[file,GraphDataString[g,
"StandardName"->PolyhedronNameToGraphName@stdname,
"AlternateStandatdNames"->PolyhedronNameToGraphName/@altnames,
"Name"->lowercasedecamel@stdname,
"AlternateNames"->lowercasedecamel/@PolyhedronNameToGraphName/@altnames,
"Information"->stdname,
"Classes"->{"UniformSkeleton"},
"Skip"->{"ConnectedInducedSubgraphPolynomial","ConnectedInducedSubgraphs","TuttePolynomial"},
TimeConstraint->10
]]]

Close [file]