Window Graph

Author

Eric W. Weisstein
July 27, 2026

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

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

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

Figure

GraphicsGrid[
Partition[
StyleGraphs[
GraphData["WindowGraph","Graph","All"]
],
UpTo[5]
],
Dividers->All
]

Source

Robertson et al. 2019

Embeddings

<<MathWorld`Graphs`

Names

GraphData["WindowGraph","StandardName"]

GraphData["WindowGraph","StandardNames"]

HouseOfGraphs

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

HouseOfGraphsLookup[hogid]

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

Wikidata

GraphData["WindowGraph","WikidataID"]

GraphData["WindowGraph","WikidataID","URL"]

Construction Robertson et al. 2019

v=DeleteCases[Join[Sequence@@Table[{i,j},{i,3},{j,3}],{{0,0},{0,4},{4,4},{4,0}}],{2,2}];
g=Graph[Range[Length[v]],UndirectedEdge@@@Sort[{{9,12},{11,12},{10,11},{9,10},{3,10},{8,11},{6,12},{1,9},{1,4},{4,6},{2,7},{3,5},{5,8},{4,5},{1,2},{2,3},{6,7},{7,8}}],VertexCoordinates->v,VertexLabels->"Name"]

CubicGraphQ[g]

r=RecognizeGraph[g]

Primary

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

All

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

CurvedEdge

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

Inexact

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

Degenerate

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

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

Embeddings

GraphDataEmbeddings[GraphData["WindowGraph"]]

Default embedding

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

g=FindCorrespondingGraph[,Join[CirclePoints[{1,Pi/2},8],CirclePoints[{1/2,Pi/2},4]]]

Configuration

GraphData["WindowGraph","Configuration"]

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

GeneralizedPetersen

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

IGraph

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

CompleteKPartite

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

Bipartite

GraphData["WindowGraph","Bipartite"]

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

Bilateral

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

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

Circular

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

Circular construction All

GraphData["WindowGraph","Hamiltonian"]

GraphData["WindowGraph","HamiltonianCycles"]

GraphData["WindowGraph","HamiltonianCycleCount"]

GraphData["WindowGraph","LCF"]

GraphData["WindowGraph","LCFSignature"]

BilateralHamiltonianCycleQ[GraphData["WindowGraph"]]

FindBilateralHamiltonianCycle[GraphData["WindowGraph"]]

Planar

GraphData["WindowGraph","Planar"]

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

MinimalCrossingNumber crossings.uos.de

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

DateString[]

Quit

MinimalCrossingNumber: "EmbeddingScheme" -> 1

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 2

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 3

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

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

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

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

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

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

MinimalCrossing

GraphData["WindowGraph","CN"]

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

GraphCrossingCount/@gs

GraphCrossingDiagram/@gs

MinimalRectilinearCrossing

GraphData["WindowGraph","RCN"]

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

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData["WindowGraph"],"CrossingNumber"->1],-4Pi/5]

GraphData["WindowGraph","V"]

e[l__]:=Sequence@@Partition[{l},2,1]

v={{0,0},{1,0},{1/4,1/3},{3/4,1/3},{1/4,2/3},{1/2,2/3},{3/4,2/3},{0,1},{1/4,1},{3/4,1},{1,1},{1/2,4/3}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,11,12,8,1,3,4,2],e[8,9,10,11],{6,12},e[3,5,9],e[4,7,10],e[5,6,7]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

MinimalCrossing construction Tutte

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

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

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

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

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

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

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

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

MinimalCrossing construction Random 1k [XXX s]

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

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

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

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

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

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

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

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

MaxMemoryUsed[]/10^9.GB

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

MinimalLocalCrossing

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

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

MinimalRectilinearLocalCrossing

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

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

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

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

Circulant

GraphData["WindowGraph","Circulant"]

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

Circulant construction

GraphData["WindowGraph","NotationRules","Circulant"]

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

CirculantGraph1@@@ci

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

CirculantGraph1@@@%

Complement graph

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

Line graph

GraphData["WindowGraph","Line"]

LineGraphQ[GraphData["WindowGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["WindowGraph"]]

Halin

GraphData["WindowGraph","Halin"]

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

Rigid

GraphData["WindowGraph","Rigid"]

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

Flexible

GraphData["WindowGraph","Flexible"]

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

Voltage

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

VoltageGraphNotation/@gs//Column

Voltage construction

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

gs=Take[%,1]

EmbeddingClasses@gs[[1]]

LCF

GraphData["WindowGraph","LCF"]

GraphData["WindowGraph","HamiltonianCycleCount"]

GraphData["WindowGraph","LCFSignature"]

GraphData["WindowGraph","LCFNotations"]

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

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

LCF compute FindBilateralHamiltonianCycle

BilateralHamiltonianCycleQ[GraphData["WindowGraph"],Debug->True]//AbsoluteTiming

(BilateralHamiltonianCycles=FindBilateralHamiltonianCycle[GraphData["WindowGraph"],Debug->True])//AbsoluteTiming

CircularEmbeddingFromHamiltonianCycle[GraphData["WindowGraph"],BilateralHamiltonianCycles[[1]]]

LCFNotation@%

LCF compute All

Divisors[GraphData["WindowGraph","V"]]

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

Tally[(lcf=LCF[GraphData["WindowGraph"],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["WindowGraph","UnitDistance"]

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

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

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

UnitDistance construction

gg=LabelGraph[]

Clear[e,g,x,y]

v={{e,-f},{c,d},{-c,-d},{i,j},{-e,f},{-g,h},{g,-h},{x,y},{a,-b},{-i,-j},{-a,b},{-x,-y}};

ee=EdgeList[gg]

eqn=UnitEdgeEquations[v,ee]//FullSimplify//Union

perpeqn=RightAngleEquation[v,{3,1,2}]

soln=Solve[Join[eqn,{g==0,j==0,perpeqn}],vars,Reals]

gs=Select[Graph[Range[Length[v]],ee,VertexCoordinates->(v/.#),VertexLabels->Automatic]&/@soln,UnitDistanceGraphEmbeddingQ]

gs={RotateGraph[,{2,5}],RotateGraph[,{2,5}],RotateGraph[,{2,5}]}

RecognizeGraph/@gs

UnitDistance by refinement of default embeddings

Graph[edge list]

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

UnitDistanceGraphEmbeddingQ[g]

Graph[vertex list, edge list]

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

UnitDistanceGraphEmbeddingQ[g]

UnitDistance by refinement of existing embeddings

gs=UnitDistanceEmbeddingRefine[Graph[VertexList[#],EdgeList[#],VertexCoordinates->GraphEmbedding[#]],MaxIterations->500]&/@Complement@@(GraphData["WindowGraph","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["WindowGraph",".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["WindowGraph"],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["WindowGraph"],"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

LabelGraph/@IndexGraph/@{,}

gs={RotateGraph[RotateGraph[,{2,6}],Pi/2],RotateGraph[,{5,1}]}

Matchstick

GraphData["WindowGraph","Matchstick"]

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

3D default

Graph3D[GraphData["WindowGraph","Edges"]]

3D construction

3D All

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

3D UnitDistance construction Random

GraphData["WindowGraph","Dimension"]

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

EdgeLengths[g]

UnitDistanceGraphEmbeddingQ[g]

3D UnitDistance

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

3D Polyhedron

GraphData["WindowGraph","Polyhedral"]

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