Jaws Graph

Author

Eric W. Weisstein
July 27, 2026

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

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

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

Figure

GraphicsGrid[
Partition[
StyleGraphs[
GraphData["JawsGraph","Graph","All"]
],
UpTo[4]
],
Dividers->All
]

Embeddings

<<MathWorld`Graphs`

Names

GraphData["JawsGraph","StandardName"]

GraphData["JawsGraph","StandardNames"]

HouseOfGraphs

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

HouseOfGraphsLookup[hogid]

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

Wikidata

GraphData["JawsGraph","WikidataID"]

GraphData["JawsGraph","WikidataID","URL"]

Primary

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

All

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

CurvedEdge

GraphData["JawsGraph","Graph","CurvedEdge"]//StyleGraphs

Inexact

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

Degenerate

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

DegenerateGraphEmbeddingTypes/@GraphData["JawsGraph","Graph","Degenerate"]//Column

Embeddings

GraphDataEmbeddings[GraphData["JawsGraph"]]

Default embedding

g0=GraphPlot[Graph[GraphData["JawsGraph","VertexList"],GraphData["JawsGraph","Edges"]]]

Configuration

GraphData["JawsGraph","Configuration"]

configurations=GraphData["JawsGraph","Graph","Configuration"]//StyleGraphs

GeneralizedPetersen

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

IGraph

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

CompleteKPartite

GraphData["JawsGraph","Graph","CompleteKPartite"]//StyleGraphs

Bipartite

GraphData["JawsGraph","Bipartite"]

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

Bilateral

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

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingTypes[#],GraphSymmetryGroup[#]}&/@GraphData["JawsGraph","Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]//Quiet

Circular

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

Circular construction All

GraphData["JawsGraph","Hamiltonian"]

GraphData["JawsGraph","HamiltonianCycles"]

GraphData["JawsGraph","HamiltonianCycleCount"]

GraphData["JawsGraph","LCF"]

GraphData["JawsGraph","LCFSignature"]

BilateralHamiltonianCycleQ[GraphData["JawsGraph"]]

FindBilateralHamiltonianCycle[GraphData["JawsGraph"]]

Planar

GraphData["JawsGraph","Planar"]

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

MinimalCrossingNumber crossings.uos.de

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

DateString[]

Quit

MinimalCrossingNumber: "EmbeddingScheme" -> 1

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 2

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 3

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

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

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

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

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

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

MinimalCrossing

GraphData["JawsGraph","CN"]

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

GraphCrossingCount/@gs

GraphCrossingDiagram/@gs

MinimalRectilinearCrossing

GraphData["JawsGraph","RCN"]

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

MinimalRectilinearCrossing construction order X [minimum RCN == ]

Length[gs=FindMinimalRectilinearCrossingGraph[GraphData["JawsGraph"],Automatic,All,"Size"->Automatic,Method->"SubgraphIsomorphism",Debug->True]]//Timing

MaxMemoryUsed[]/10^9. GB

Tally[RectilinearCrossingCount/@gs]//Timing

Tally[RectilinearLocalCrossingCount/@gs]//Timing

GraphSymmetryGroup/@gs//Tally

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

Take[gs,UpTo[100]]

MinimalRectilinearCrossing construction order X [minimum RCN == ]

Length[gs=FindMinimalRectilinearCrossingGraph[GraphData["JawsGraph"],Automatic,All,"Size"->XXX,Method->"SubgraphIsomorphism",Debug->True]]//Timing

MaxMemoryUsed[]/10^9. GB

Tally[RectilinearCrossingCount/@gs]//Timing

Tally[RectilinearLocalCrossingCount/@gs]//Timing

GraphSymmetryGroup/@gs//Tally

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

Take[gs,UpTo[100]]

MinimalCrossing construction

g=RotateGraph[MinimalCrossingGraph[GraphData["JawsGraph"],"CrossingNumber"->2]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

MinimalCrossing construction Tutte

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

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

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

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

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

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

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

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

MinimalCrossing construction Random 1k [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["JawsGraph"],
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["JawsGraph"],TimeConstraint->1])//AbsoluteTiming

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["JawsGraph"],
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["JawsGraph"],TimeConstraint->1])//AbsoluteTiming

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["JawsGraph"],
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["JawsGraph"],TimeConstraint->1])//AbsoluteTiming

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

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["JawsGraph","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["JawsGraph","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["JawsGraph"],TimeConstraint->1])//AbsoluteTiming

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

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["JawsGraph","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["JawsGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@RandomGraphs;

MaxMemoryUsed[]/10^9.GB

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

MinimalLocalCrossing

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

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

MinimalRectilinearLocalCrossing

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

GraphData["JawsGraph","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["JawsGraph"],TimeConstraint->10])//AbsoluteTiming

(gs=Select[Table[MinimalCrossingGraph[GraphData["JawsGraph"],"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["JawsGraph"],TimeConstraint->10])//AbsoluteTiming

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

Circulant

GraphData["JawsGraph","Circulant"]

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

Circulant construction

GraphData["JawsGraph","NotationRules","Circulant"]

(ci=CirculantIndices[GraphData["JawsGraph"]])//AbsoluteTiming

CirculantGraph1@@@ci

Cases[GraphData["JawsGraph","StandardNames"],{"Circulant",i_}:>i]

CirculantGraph1@@@%

Complement graph

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

Line graph

GraphData["JawsGraph","Line"]

LineGraphQ[GraphData["JawsGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["JawsGraph"]]

Halin

GraphData["JawsGraph","Halin"]

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

Rigid

GraphData["JawsGraph","Rigid"]

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

Flexible

GraphData["JawsGraph","Flexible"]

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

Voltage

gs=GraphData["JawsGraph","Graph","Voltage"]//StyleGraphs

VoltageGraphNotation/@gs//Column

Voltage construction

VoltageGraph[GraphData["JawsGraph"],"MaxResults"->20,Debug->True]

LCF

GraphData["JawsGraph","LCF"]

GraphData["JawsGraph","HamiltonianCycleCount"]

GraphData["JawsGraph","LCFSignature"]

GraphData["JawsGraph","LCFNotations"]

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

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

LCF compute All

Divisors[GraphData["JawsGraph","V"]]

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

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

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

Length[gs=nicelcfgs]

gs

{,,}

NiceLCFEmbeddingsWithSymmetryGroups={#,GraphSymmetryGroup[#]}&/@gs;

DeleteCases[NiceLCFEmbeddingsWithSymmetryGroups,{_,TransformationGroup[{Reflection[InfiniteLine[{0,0},{0,-1}]]}]}]

UnitDistance

GraphData["JawsGraph","UnitDistance"]

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

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

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

UnitDistance by refinement of default embeddings

Graph[edge list]

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

UnitDistanceGraphEmbeddingQ[g]

Graph[vertex list, edge list]

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

UnitDistanceGraphEmbeddingQ[g]

UnitDistance by refinement of existing embeddings

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

TextGrid[
gdata=SortBy[
{#,DegenerateGraphEmbeddingQ[#],GraphSymmetryGroup[#],Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
],
Dividers->All
]

Cases[gdata,{__,0.}]

Symmetric:

gs=Cases[
gdata,
{g_,_,sym:Except[TransformationGroup[{}]],diff_}:>{
HighlightGraph[g1=BilateralGraph[CenterGraph[g],"ReturnNonbilateralGraph"->True],Flatten[FindCoincidentVertices[g1]]],
sym,
diff
}]

UnitDistance by refinement of stored LCF embeddings

FileByteCount[file=FileNameJoin[{$PropertiesDir,"LCFNotations-Nice",GraphNameToFileName["JawsGraph",".m.gz"]}]]

Tally[(lcf=Import[file])[[All,-1]]]//Timing//AbsoluteTiming

lcf=LCFNotation/@nicelcfgs;

gs=UnitDistanceEmbeddingRefine[LCFGraph@@#,MaxIterations->500]&/@lcf;//Timing

gdata=SortBy[
{#,Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
]

Cases[gdata,{g_?UnitDistanceGraphEmbeddingQ,_}:>g]

NondegenerateSymmetricGraphs=Select[
gdata,
And[
DegenerateGraphEmbeddingQ[#[[1]]]===False,
GraphSymmetryGroup[#[[1]]]=!=TransformationGroup[{}]
]&
]/.g_Graph:>CenterGraph[g]

UnitDistance by FindUnitDistanceEmbedding  (random embeddings returning on messages)

gs=FindUnitDistanceEmbedding[GraphData["JawsGraph"],50,"ReturnWithMessage"->True];//Quiet//Timing

gdata=SortBy[
{#,Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
]

Cases[gdata,{g_?UnitDistanceGraphEmbeddingQ,_}:>g]

SymmetricEmebddings=Select[gdata,GraphSymmetryGroup[#[[1]]]=!=TransformationGroup[{}]&]

UnitDistance by FindUnitDistanceEmbedding  (random embedding waiting until UD is found)

FindUnitDistanceEmbedding[GraphData["JawsGraph"],"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

g=RotateGraph[LabelGraph@IndexGraph@,{2,19}]

gs={,};

Matchstick

GraphData["JawsGraph","Matchstick"]

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

3D default

Graph3D[GraphData["JawsGraph","Edges"]]

3D construction

3D All

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

3D UnitDistance construction Random

GraphData["JawsGraph","Dimension"]

(g=UnitDistanceEmbeddingRefine[
Graph3D[
GraphData["JawsGraph","VertexList"],
GraphData["JawsGraph","EdgeList"]
],
WorkingPrecision->25])//Timing

EdgeLengths[g]

UnitDistanceGraphEmbeddingQ[g]

3D UnitDistance

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

3D Polyhedron

GraphData["JawsGraph","Polyhedral"]

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

Construction

Robertson et al. 2019

v=Join[Table[{i,2},{i,0,7}],{#,1}&/@{1,3,4,6},Table[{i,0},{i,0,7}]];
jaws=Graph[
Range[Length[v]],
UndirectedEdge@@@Sort[Join[Partition[Range[8],2,1],Partition[Range[13,20],2,1],{{1,13},{8,20},{4,10},{10,16},{7,12},{12,19},{9,10},{11,12},{13,18},{15,20},{1,6},{3,8},{9,14},{2,9},{11,17},{5,11}}]],
VertexCoordinates->v,
VertexLabels->"Name"
]

RecognizeGraph[g]

VertexCount[g]

CubicGraphQ[g]

CrossingNumber[g,MaxIterations->10^2]//AbsoluteTiming

AlmostPlanarGraphQ[g]//Timing

StringReplace[ExportString[g,"Graph6"],">>graph6<<"|"\n"->""]

MinimalRectilinearCrossing

g=MinimalCrossingGraph[g,"CrossingNumber"->2]

quads=AdjacencyList[g,#]&/@{21,22}

Pairs[l_]:=Module[{twos={l[[1]],#}&/@Rest[l]},
{#,Complement[l,#]}&/@twos
]

p1=Pairs[quads[[1]]]

p2=Pairs[quads[[2]]]

combs=Flatten[#,1]&/@Flatten[Outer[List,p1,p2,1],1]

Length[combs]

g0=Graph[Range[20],Select[EdgeList[g],FreeQ[#,21|22]&],VertexCoordinates->Take[GraphEmbedding[g],20]]

gs=EdgeAdd[g0,UndirectedEdge@@@#]&/@combs

GraphPlot/@Pick[gs,IsomorphicGraphQ[#,jaws]&/@gs]

Rotate[,2.1]

v={#/5,#2/6}&@@@{{0,0},{5,0},{5,6},{0,6},{1,1},{1,3},{1,5},{2,1},{2,2},{2,3},{2,4},{2,5},{3,1},{3,2},{3,3},{3,4},{3,5},{4,1},{4,3},{4,5}};
g=Graph[Range[Length[v]],UndirectedEdge@@@Sort[{{1,13},{1,5},{4,7},{3,20},{2,18},{4,17},{17,20},{3,12},{2,8},{5,6},{6,7},{18,19},{19,20},{13,14},{14,15},{15,16},{7,12},{16,17},{8,9},{9,10},{10,11},{11,12},{11,16},{9,14},{5,8},{13,18},{6,10},{15,19},{1,4},{2,3}}],VertexCoordinates->v,VertexLabels->"Name"
]

IsomorphicGraphQ[g,jaws]

RectilinearCrossingCount@g

ToCommonEdges[g,jaws]

GraphData

GraphDataClassString

GraphDataClassString["JawsGraph"]

GraphDataString

file=OpenWrite["~/jaws.m"]

WriteString[file,GraphDataString[g,
"StandardName"->"JawsGraph",
"Name"->"jaws graph",
"EmbeddingClasses"->{{"Degenerate","Primary"}},
TimeConstraint->10,
Debug->True,
"Skip"->{"Arboricity"}
]]

Close[file]