Dodecadodecahedral Graph

Author

Eric W. Weisstein
September 5, 2023

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

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Graphics

GraphicsRow[GraphData["DodecadodecahedralGraph","Graph","Bilateral"]//StyleGraphs]

Embeddings

<<MathWorld`Graphs`

Names

GraphData["DodecadodecahedralGraph","StandardName"]

GraphData["DodecadodecahedralGraph","StandardNames"]

Primary

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

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["DodecadodecahedralGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

Bilateral

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

Circular

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

Planar

GraphData["DodecadodecahedralGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

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

CrossingNumber[GraphData["DodecadodecahedralGraph"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["DodecadodecahedralGraph"],"CrossingNumber"->cn]]

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

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["DodecadodecahedralGraph","Circulant"]

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

Halin

GraphData["DodecadodecahedralGraph","Halin"]

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

LCF

GraphData["DodecadodecahedralGraph","LCF"]

GraphData["DodecadodecahedralGraph","HamiltonianCycleCount"]

GraphData["DodecadodecahedralGraph","LCFSignature"]

GraphData["DodecadodecahedralGraph","LCFNotations"]

%/.Missing[_,l_]:>Tally[l[[All,-1]]]

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

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

LCF compute all

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

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

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[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,XXX}]]}])//Timing

Length[gs]

BilateralGraphEmbeddingQ/@gs

UnitDistance

GraphData["DodecadodecahedralGraph","UnitDistance"]

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

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

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

UnitDistance by refinement

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

Select[%,UnitDistanceGraphEmbeddingQ]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

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

Matchstick

GraphData["DodecadodecahedralGraph","Matchstick"]

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

3D default

Graph3D[GraphData["DodecadodecahedralGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D construction from uniform polyhedra

BulatovUniformName/@{41,70,67}

gs=PolyhedronGraph[p=BulatovUniformPolyhedron[#],Mean[PolyhedronEdgeLengths[p]]]&/@{41,70,67}

RecognizeGraph/@gs

UnitDistanceGraphEmbeddingQ/@gs

g=gs[[1]]

RecognizeGraph@g

v=GraphEmbedding[g]

vexact=Map[RootApproximant[#,2]&,Chop@v,{2}]

Abs[v-vexact]//Max

g=Annotate[g,VertexCoordinates->vexact]

EdgeLengths[g]//RootReduce//Union

3D Polyhedron

GraphData["DodecadodecahedralGraph","Polyhedral"]

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

Properties

PolyhedralEmbeddings

GraphData["DodecadodecahedralGraph","PolyhedralEmbeddings"]

GraphData

GraphDataClassString["DodecadodecahedralGraph"]