Ditrigonal Icosidodecahedral Graph

Author

Eric W. Weisstein
September 4, 2023

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

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

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

Graphics

GraphicsRow@Join[
GraphData["DitrigonalIcosidodecahedralGraph","Graph","All"][[{1,-1}]],
Select[GraphData["DitrigonalIcosidodecahedralGraph","Graph","All"],BilateralGraphEmbeddingQ]]//StyleGraphs

Embeddings

<<MathWorld`Graphs`

Names

GraphData["DitrigonalIcosidodecahedralGraph","StandardName"]

GraphData["DitrigonalIcosidodecahedralGraph","StandardNames"]

Primary

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

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["DitrigonalIcosidodecahedralGraph","EdgeList"]]

Default embedding

g=Graph@GraphData["DitrigonalIcosidodecahedralGraph","Edges"]

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

RecognizeGraph[g]

Degenerate

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

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

Perspective

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

Perspective construction

PolyhedronGraph[GraphData["DitrigonalIcosidodecahedralGraph","Polyhedron"]]

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

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

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

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

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

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

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

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

Bilateral

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

Circular

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

Planar

GraphData["DitrigonalIcosidodecahedralGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

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

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

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["DitrigonalIcosidodecahedralGraph","Circulant"]

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

Line graph

GraphData["DitrigonalIcosidodecahedralGraph","Line"]

LineGraphQ[GraphData["DitrigonalIcosidodecahedralGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["DitrigonalIcosidodecahedralGraph"]]

Halin

GraphData["DitrigonalIcosidodecahedralGraph","Halin"]

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

Rigid

GraphData["DitrigonalIcosidodecahedralGraph","Rigid"]

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

Flexible

GraphData["DitrigonalIcosidodecahedralGraph","Flexible"]

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

LCF

GraphData["DitrigonalIcosidodecahedralGraph","LCF"]

GraphData["DitrigonalIcosidodecahedralGraph","HamiltonianCycleCount"]

GraphData["DitrigonalIcosidodecahedralGraph","LCFSignature"]

GraphData["DitrigonalIcosidodecahedralGraph","LCFNotations"]

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

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

LCF compute all

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

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

MaxMemoryUsed[]/10.^9 "GB"

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

DeleteCases[lcf,{_,1}]

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]

UnitDistance

GraphData["DitrigonalIcosidodecahedralGraph","UnitDistance"]

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

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

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

UnitDistance by refinement

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

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

LabelGraph/@{,,,}

g=;

Matchstick

GraphData["DitrigonalIcosidodecahedralGraph","Matchstick"]

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

3D default

Graph3D[GraphData["DitrigonalIcosidodecahedralGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D construction from uniform polyhedra

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["DitrigonalIcosidodecahedralGraph","Polyhedral"]

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

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

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

GraphData

GraphDataClassString

GraphDataClassString["DitrigonalIcosidodecahedralGraph"]