Reye Graph

Author

Eric W. Weisstein
September 26, 2024

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

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

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

Definition

Levi graph of:
*16 lines and the same 12 points such that four lines pass through every point and three points lie on every line
* 12 planes and 12 points such that six points lie in every plane and six planes pass through every point

Properties

GraphData["ReyeGraph",{"V","E"}]

Embeddings

<<MathWorld`Graphs`

Names

GraphData["ReyeGraph","StandardName"]

GraphData["ReyeGraph","StandardNames"]

Primary

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

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["ReyeGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Incidence

GraphData["ReyeGraph","Incidence"]

Select[GraphData[All],GraphData[#,"LeviGraph","Name"]==="ReyeGraph"&]

lines=GraphData["ReyeGraph","IncidenceLines"]

levi=LeviGraph[lines]

RecognizeGraph@levi

Perspective

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

Bilateral

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

Circular

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

Planar

GraphData["ReyeGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

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

Table[MinimalCrossingGraph[GraphData["ReyeGraph"],"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["ReyeGraph","Circulant"]

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

Complement graph

GraphData["ReyeGraph","ComplementGraph","Name"]

Line graph

GraphData["ReyeGraph","Line"]

LineGraphQ[GraphData["ReyeGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["ReyeGraph"]]

Halin

GraphData["ReyeGraph","Halin"]

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

Rigid

GraphData["ReyeGraph","Rigid"]

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

Flexible

GraphData["ReyeGraph","Flexible"]

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

LCF

GraphData["ReyeGraph","LCF"]

GraphData["ReyeGraph","LCFSignature"]

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

UnitDistance

GraphData["ReyeGraph","UnitDistance"]

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

Matchstick

GraphData["ReyeGraph","Matchstick"]

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

MinimalCrossing

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

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

Default embedding

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

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

RecognizeGraph[g]

MinimalCrossing construction

CrossingNumber[GraphData["ReyeGraph"],TimeConstraint->10]//AbsoluteTiming

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

GraphPlot[MinimalCrossingGraph[GraphData["ReyeGraph"],"CrossingNumber"->20]]

RecognizeGraph[g]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData["ReyeGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["ReyeGraph","Polyhedral"]

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