Concertina Graph

Author

Eric W. Weisstein
July 27, 2026

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

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

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

Figure

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

Embeddings

<<MathWorld`Graphs`

Names

GraphData["ConcertinaGraph","StandardName"]

GraphData["ConcertinaGraph","StandardNames"]

HouseOfGraphs

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

HouseOfGraphsLookup[hogid]

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

Wikidata

GraphData["ConcertinaGraph","WikidataID"]

GraphData["ConcertinaGraph","WikidataID","URL"]

Primary

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

All

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

CurvedEdge

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

Robertson et al. 2019

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

CubicGraphQ[g]

RecognizeGraph[g]

Inexact

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

Degenerate

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

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

Embedding types

GraphDataEmbeddings[GraphData["ConcertinaGraph","EdgeList"]]

Default embedding

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

From Ed Pegg, Jr., Nov 11, 2022:

g=Graph[GraphData["ConcertinaGraph"],VertexCoordinates->Thread[{3,6,2,1,5,10,7,4,8,11,12,16,9,18,15,14,17,13}->{{1,0},{2,0},{3,0},{0,1},{1,1},{2,1},{3,1},{4,1},{1,2},{3,2},{0,3},{1,3},{2,3},{3,3},{4,3},{1,4},{2,4},{3,4}}]]

RecognizeGraph[g]

Configuration

GraphData["ConcertinaGraph","Configuration"]

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

GeneralizedPetersen

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

IGraph

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

CompleteKPartite

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

Bipartite

GraphData["ConcertinaGraph","Bipartite"]

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

Bilateral

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

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

Circular

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

Circular construction All

GraphData["ConcertinaGraph","Hamiltonian"]

GraphData["ConcertinaGraph","HamiltonianCycles"]

GraphData["ConcertinaGraph","HamiltonianCycleCount"]

GraphData["ConcertinaGraph","LCF"]

GraphData["ConcertinaGraph","LCFSignature"]

BilateralHamiltonianCycleQ[GraphData["ConcertinaGraph"]]

FindBilateralHamiltonianCycle[GraphData["ConcertinaGraph"]]

Planar

GraphData["ConcertinaGraph","Planar"]

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

MinimalCrossingNumber crossings.uos.de [verified 2]

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

DateString[]

MinimalCrossingNumber: "EmbeddingScheme" -> 1

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 2

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 3

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

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

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

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

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

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

MinimalCrossing

GraphData["ConcertinaGraph","CN"]

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

GraphCrossingCount/@gs

GraphCrossingDiagram/@gs

MinimalRectilinearCrossing

GraphData["ConcertinaGraph","RCN"]

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

MinimalCrossing construction Tutte

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

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

MinimalCrossing construction (manual 1)

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData["ConcertinaGraph"],"CrossingNumber"->-1],Pi/6]

v=Delete[Join[Flatten[Table[{i,j},{i,-1,1},{j,-2,2}],1],{{-3,-3},{3,-3},{3,3},{-3,3}}],{8}];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[15,16,10,6,1,15,4,3,2,18,17,11,12,13,16],e[4,5,9,14,13],{1,2},{10,11},e[6,7,8,9],{3,8},{7,12},{5,18},{14,17}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

MinimalRectilinearCrossing construction (manual)

mc=MinimalCrossingGraph[GraphData["ConcertinaGraph"],"CrossingNumber"->2]

mc2=VertexDelete[mc,{19,20}]

d=VertexDegree[mc2]

add=Flatten[MapIndexed[Table[#2[[1]],#]&,3-d]]

Length[sub=Select[Subsets[Subsets[add,{2}],{Length[add]/2}],Intersection[EdgeList[mc2],#]==={}&]]

gs=SetProperty[EdgeAdd[mc2,UndirectedEdge@@@#],{VertexLabels->None,VertexCoordinates->Take[GraphEmbedding[mc],18]}]&/@sub;

SelectFirst[gs,IsomorphicGraphQ[#,GraphData["ConcertinaGraph"]]&]

GraphPlot[%]

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

VertexDegree[g]

r=RecognizeGraph[g]

rcn=RectilinearCrossingCount@g

vs=ToCommonEdges[g,GraphData[r]]

MinimalCrossing construction Random 1k [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["ConcertinaGraph"],
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

<<MathWorld`Graphs`

GraphPlot@StraightenGraph@RotateGraph[,.5]

MinimalCrossing construction Random 5k [XXX s]

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

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

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

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

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

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

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

MaxMemoryUsed[]/10^9.GB

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

MinimalLocalCrossing

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

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

MinimalRectilinearLocalCrossing

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

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

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

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

Circulant

GraphData["ConcertinaGraph","Circulant"]

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

Circulant construction

GraphData["ConcertinaGraph","NotationRules","Circulant"]

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

CirculantGraph1@@@ci

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

CirculantGraph1@@@%

Complement graph

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

Line graph

GraphData["ConcertinaGraph","Line"]

LineGraphQ[GraphData["ConcertinaGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["ConcertinaGraph"]]

Halin

GraphData["ConcertinaGraph","Halin"]

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

Rigid

GraphData["ConcertinaGraph","Rigid"]

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

Flexible

GraphData["ConcertinaGraph","Flexible"]

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

Voltage

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

VoltageGraphNotation/@gs//Column

Voltage construction

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

gs=Take[%,XXX]

LCF

GraphData["ConcertinaGraph","LCF"]

GraphData["ConcertinaGraph","HamiltonianCycleCount"]

GraphData["ConcertinaGraph","LCFSignature"]

GraphData["ConcertinaGraph","LCFNotations"]

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

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

LCF compute FindBilateralHamiltonianCycle

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

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

LCF compute All

Divisors[GraphData["ConcertinaGraph","V"]]

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

Tally[(lcf=LCF[GraphData["ConcertinaGraph"],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

UnitDistance

GraphData["ConcertinaGraph","UnitDistance"]

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

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

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

UnitDistance by refinement of default embeddings

Graph[edge list]

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

UnitDistanceGraphEmbeddingQ[g]

Graph[vertex list, edge list]

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

UnitDistanceGraphEmbeddingQ[g]

UnitDistance by refinement of existing embeddings

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

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

gs={,,,,,,,,,,,,,,,}

gs={,};

Matchstick

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

3D default

Graph3D[GraphData["ConcertinaGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

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

GraphData

GraphDataString

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

WriteString[file,GraphDataString[g,
"StandardName"->"ConcertinaGraph",
"Name"->"concertina graph",
TimeConstraint->10,
Debug->True,
"Skip"->{"Arboricity"}
]]

Close[file]