Planar Hypotraceable Graph

Author

Eric W. Weisstein
June 1, 2022

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

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Graphics

Grid[planarhypotraceable=SortBy[GraphData[{"Hypotraceable","Planar"},{"V","#"}],First],Dividers->All,Alignment->{{Decimal,Left}}]

GraphicsRow[Show[GraphData[#]//StyleGraphs[#,VertexSize->{"Scaled",.025}]&,PlotLabel->GraphData[#,"Name"],LabelStyle->{14,FontFamily->"Times",Black,Italic}]&/@planarhypotraceable[[All,-1]],ImageSize->600,Dividers->All]

Embeddings

{"PlanarHypotraceable", 162}

GraphData[ {"PlanarHypotraceable",162},"StandardName"]

All

GraphData[ {"PlanarHypotraceable",162},"Graph","All"]//StyleGraphs

Halin

GraphData[ {"PlanarHypotraceable",162},"Graph","Halin"]//StyleGraphs

LCF

GraphData[ {"PlanarHypotraceable",162},"LCFSignature"]

GraphData[ {"PlanarHypotraceable",162},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[ {"PlanarHypotraceable",162},"UnitDistance"]

UnitDistanceGraphQ[GraphData[ {"PlanarHypotraceable",162}],Debug->True]//Timing

GraphData[ {"PlanarHypotraceable",162},"Graph","UnitDistance"]//StyleGraphs

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

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[ {"PlanarHypotraceable",162},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[ {"PlanarHypotraceable",162}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{ {"PlanarHypotraceable",162},l=FindSubgraphUnitDistanceEmbedding[GraphData[ {"PlanarHypotraceable",162}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[ {"PlanarHypotraceable",162},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[ {"PlanarHypotraceable",162},{"RCN","CN"}]

GraphData[ {"PlanarHypotraceable",162},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[ {"PlanarHypotraceable",162},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[ {"PlanarHypotraceable",162}],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[ {"PlanarHypotraceable",162},"Edges"]]

3D All

GraphData[ {"PlanarHypotraceable",162},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[ {"PlanarHypotraceable",162},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[ {"PlanarHypotraceable",162},"Polyhedral"]

Construction

Wiener Gabor
Thu, 21 May 2009 06:14:17

g162=FromAdjacencyLists[{{2,4,5,6},{1,3,7,8},{2,4,9,10},{1,3,11,12},{1,16,17},{1,13,18},{2,13,19},{2,14,20},{3,14,21},{3,15,22},{4,15,23},{4,16,24},{6,7,26},{8,9,27},{10,11,28},{5,12,25},{5,18,29},{6,17,30},{7,20,161},{8,19,31},{9,22,32},{10,21,33},{11,24,34},{12,23,35},{16,29,35},{13,30,161},{14,31,32},{15,33,34},{17,25,36},{18,26,36},{20,27,37},{21,27,38},{22,28,38},{23,28,39,40},{24,25,39},{29,30,39,116},{31,38,161},{32,33,37,40},{34,35,36},{34,38,120},{42,44,45,46},{41,43,47,48},{42,44,49,50},{41,43,51,52},{41,56,57},{41,53,58},{42,53,59},{42,54,60},{43,54,61},{43,55,62},{44,55,63},{44,56,64},{46,47,66},{48,49,67},{50,51,68},{45,52,65},{45,58,69},{46,57,70},{47,60,161},{48,59,71},{49,62,72},{50,61,73},{51,64,74},{52,63,75},{56,69,75},{53,70,161},{54,71,72},{55,73,74},{57,65,76},{58,66,76},{60,67,77},{61,67,78},{62,68,78},{63,68,79,80},{64,65,79},{69,70,79,156},{71,78,161},{72,73,77,80},{74,75,76},{74,78,160},{82,84,85,86},{81,83,87,88},{82,84,89,90},{81,83,91,92},{81,96,97},{81,93,98},{82,93,99},{82,94,100},{83,94,101},{83,95,102},{84,95,103},{84,96,104},{86,87,106},{88,89,107},{90,91,108},{85,92,105},{85,98,109},{86,97,110},{87,100,162},{88,99,111},{89,102,112},{90,101,113},{91,104,114},{92,103,115},{96,109,115},{93,110,162},{94,111,112},{95,113,114},{97,105,116},{98,106,116},{100,107,117},{101,107,118},{102,108,118},{103,108,119,120},{104,105,119},{36,109,110,119},{111,118,162},{112,113,117,120},{114,115,116},{40,114,118},{122,124,125,126},{121,123,127,128},{122,124,129,130},{121,123,131,132},{121,136,137},{121,133,138},{122,133,139},{122,134,140},{123,134,141},{123,135,142},{124,135,143},{124,136,144},{126,127,146},{128,129,147},{130,131,148},{125,132,145},{125,138,149},{126,137,150},{127,140,162},{128,139,151},{129,142,152},{130,141,153},{131,144,154},{132,143,155},{136,149,155},{133,150,162},{134,151,152},{135,153,154},{137,145,156},{138,146,156},{140,147,157},{141,147,158},{142,148,158},{143,148,159,160},{144,145,159},{76,149,150,159},{151,158,162},{152,153,157,160},{154,155,156},{80,154,158},{19,26,37,59,66,77},{99,106,117,139,146,157}}];

GraphPlot[g162]

g2=LinesToGraph[]

ToCommonEdges[g2,Graph[{"PlanarHypotraceable",162}]]//Round

RecognizeGraph[g2]

FastIsomorphicQ[g2,Graph[{"PlanarHypotraceable",162}]]

PlanarQ[g162]//Timing

FindHamiltonianCycle[g162]//Timing

FastHamiltonianQ[g162]//Timing

HypohamiltonianQ[g162,False]//Timing

TraceableQ[g162,{}]//Timing

HypotraceableQ[g162]//Timing

{"PlanarHypotraceable", 186}

GraphData[{"PlanarHypotraceable",186},"StandardName"]

All

GraphData[{"PlanarHypotraceable",186},"Graph","All"]//StyleGraphs

Halin

GraphData[{"PlanarHypotraceable",186},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"PlanarHypotraceable",186},"LCFSignature"]

GraphData[{"PlanarHypotraceable",186},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"PlanarHypotraceable",186},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"PlanarHypotraceable",186}],Debug->True]//Timing

GraphData[{"PlanarHypotraceable",186},"Graph","UnitDistance"]//StyleGraphs

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

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"PlanarHypotraceable",186},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"PlanarHypotraceable",186}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"PlanarHypotraceable",186},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"PlanarHypotraceable",186}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"PlanarHypotraceable",186},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"PlanarHypotraceable",186},{"RCN","CN"}]

GraphData[{"PlanarHypotraceable",186},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"PlanarHypotraceable",186},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"PlanarHypotraceable",186}],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"PlanarHypotraceable",186},"Edges"]]

3D All

GraphData[{"PlanarHypotraceable",186},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"PlanarHypotraceable",186},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"PlanarHypotraceable",186},"Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData[{"PlanarHypotraceable",186},"Polyhedron"]}]

Construction from 4 Zamfirescu Graphs

Wiener Gabor
Thu, 21 May 2009 06:14:17

zam47=DeleteVertex[Graph["ZamfirescuGraph48"],48]

g186=Contract[Contract[AddEdges[Graph[List/@(Join@@Table[Edges[zam47]+47n,{n,0,3}]),List/@(Join@@Table[2Through[{Cos,Sin}[n π/2]]+#&/@Vertices[zam47],{n,0,3}])],{{32,126},{47,141},{79,173},{94,188}}],{1,48}],{93,140}]

g186=Graph[{{{33, 185}}, {{36, 185}}, {{44, 185}}, {{1, 2}}, {{1, 8}}, {{1, 9}}, {{2, 3}}, {{2, 11}}, {{3, 4}}, {{3, 13}}, {{4, 5}}, 
  {{4, 15}}, {{5, 6}}, {{5, 17}}, {{6, 7}}, {{6, 19}}, {{7, 8}}, {{7, 21}}, {{8, 23}}, {{9, 10}}, {{9, 24}}, {{9, 34}}, 
  {{10, 11}}, {{10, 25}}, {{11, 12}}, {{12, 13}}, {{12, 26}}, {{13, 14}}, {{13, 38}}, {{14, 15}}, {{14, 27}}, {{15, 16}}, 
  {{16, 17}}, {{16, 28}}, {{17, 18}}, {{17, 42}}, {{18, 19}}, {{18, 29}}, {{19, 20}}, {{20, 21}}, {{20, 30}}, {{21, 22}}, 
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  {{45, 46}}, {{79, 185}}, {{82, 185}}, {{90, 185}}, {{47, 48}}, {{47, 54}}, {{47, 55}}, {{48, 49}}, {{48, 57}}, {{49, 50}}, 
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  {{88, 89}}, {{89, 90}}, {{90, 91}}, {{91, 92}}, {{125, 186}}, {{128, 186}}, {{136, 186}}, {{93, 94}}, {{93, 100}}, 
  {{93, 101}}, {{94, 95}}, {{94, 103}}, {{95, 96}}, {{95, 105}}, {{96, 97}}, {{96, 107}}, {{97, 98}}, {{97, 109}}, {{98, 99}}, 
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  {{119, 131}}, {{120, 133}}, {{121, 122}}, {{121, 135}}, {{122, 137}}, {{123, 124}}, {{124, 125}}, {{125, 126}}, {{126, 127}}, 
  {{127, 128}}, {{128, 129}}, {{128, 132}}, {{129, 130}}, {{130, 131}}, {{131, 132}}, {{132, 133}}, {{132, 136}}, {{133, 134}}, 
  {{134, 135}}, {{135, 136}}, {{136, 137}}, {{137, 138}}, {{171, 186}}, {{174, 186}}, {{182, 186}}, {{139, 140}}, {{139, 146}}, 
  {{139, 147}}, {{140, 141}}, {{140, 149}}, {{141, 142}}, {{141, 151}}, {{142, 143}}, {{142, 153}}, {{143, 144}}, {{143, 155}}, 
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  {{165, 166}}, {{165, 177}}, {{166, 179}}, {{167, 168}}, {{167, 181}}, {{168, 183}}, {{169, 170}}, {{170, 171}}, {{171, 172}}, 
  {{172, 173}}, {{173, 174}}, {{174, 175}}, {{174, 178}}, {{175, 176}}, {{176, 177}}, {{177, 178}}, {{178, 179}}, {{178, 182}}, 
  {{179, 180}}, {{180, 181}}, {{181, 182}}, {{182, 183}}, {{183, 184}}, {{31, 123}}, {{46, 138}}, {{77, 169}}, {{92, 184}}}, 
 {{{3.4649, -0.3987}}, {{3.5067, -0.2976}}, {{3.6075999999999997, -0.2557}}, {{3.7085, -0.2976}}, {{3.7503, -0.3987}}, 
  {{3.7085, -0.4998}}, {{3.6075999999999997, -0.5416}}, {{3.5067, -0.4998}}, {{3.3222, -0.3987}}, {{3.3439, -0.2892}}, 
  {{3.4058, -0.1965}}, {{3.4984, -0.1345}}, {{3.6075999999999997, -0.1127}}, {{3.7168, -0.1345}}, {{3.8094, -0.1965}}, 
  {{3.8712, -0.2892}}, {{3.8929, -0.3987}}, {{3.8712, -0.5081}}, {{3.8094, -0.6009}}, {{3.7168, -0.6629}}, 
  {{3.6075999999999997, -0.6846}}, {{3.4984, -0.6629}}, {{3.4058, -0.6009}}, {{3.3439, -0.5081}}, 
  {{3.2171000000000003, -0.1023}}, {{3.3118, -0.0074}}, {{3.9034, -0.0074}}, {{3.998, -0.1023}}, {{3.998, -0.6951}}, 
  {{3.9034, -0.79}}, {{3.3118, -0.79}}, {{3.2171000000000003, -0.6951}}, {{3.0978000000000003, -0.6341}}, {{3.1258, -0.3987}}, 
  {{3.0978000000000003, -0.1632}}, {{2.9655, 0.2448}}, {{3.3726000000000003, 0.1122}}, {{3.6075999999999997, 0.0841}}, 
  {{3.8426, 0.1122}}, {{4.2496, 0.2448}}, {{4.1174, -0.1632}}, {{4.0893999999999995, -0.3987}}, {{4.1174, -0.6341}}, 
  {{4.2496, -1.0421}}, {{3.8426, -0.9095}}, {{3.6075999999999997, -0.8815}}, {{1.4649, 1.6013}}, {{1.5067, 1.7024}}, 
  {{1.6076, 1.7443}}, {{1.7085, 1.7024}}, {{1.7503, 1.6013}}, {{1.7085, 1.5002}}, {{1.6076, 1.4584000000000001}}, 
  {{1.5067, 1.5002}}, {{1.3222, 1.6013}}, {{1.3439, 1.7107999999999999}}, {{1.4058, 1.8035}}, {{1.4984, 1.8655}}, 
  {{1.6076, 1.8873}}, {{1.7168, 1.8655}}, {{1.8094, 1.8035}}, {{1.8712, 1.7107999999999999}}, {{1.8929, 1.6013}}, 
  {{1.8712, 1.4919}}, {{1.8094, 1.3991}}, {{1.7168, 1.3371}}, {{1.6076, 1.3154}}, {{1.4984, 1.3371}}, {{1.4058, 1.3991}}, 
  {{1.3439, 1.4919}}, {{1.2171, 1.8977}}, {{1.3118, 1.9926}}, {{1.9034, 1.9926}}, {{1.998, 1.8977}}, {{1.998, 1.3049}}, 
  {{1.9034, 1.21}}, {{1.3118, 1.21}}, {{1.2171, 1.3049}}, {{1.0978, 1.3659}}, {{1.1258, 1.6013}}, {{1.0978, 1.8368}}, 
  {{0.9655, 2.2448}}, {{1.3726, 2.1122}}, {{1.6076, 2.0841}}, {{1.8426, 2.1122}}, {{2.2496, 2.2448}}, {{2.1174, 1.8368}}, 
  {{2.0894, 1.6013}}, {{2.1174, 1.3659}}, {{2.2496, 0.9579}}, {{1.8426, 1.0905}}, {{1.6076, 1.1185}}, 
  {{-0.5350999999999999, -0.3987}}, {{-0.49330000000000007, -0.2976}}, {{-0.3924000000000001, -0.2557}}, 
  {{-0.2915000000000001, -0.2976}}, {{-0.24970000000000003, -0.3987}}, {{-0.2915000000000001, -0.4998}}, 
  {{-0.3924000000000001, -0.5416}}, {{-0.49330000000000007, -0.4998}}, {{-0.6778, -0.3987}}, {{-0.6560999999999999, -0.2892}}, 
  {{-0.5942000000000001, -0.1965}}, {{-0.5016, -0.1345}}, {{-0.3924000000000001, -0.1127}}, {{-0.2831999999999999, -0.1345}}, 
  {{-0.1906000000000001, -0.1965}}, {{-0.12880000000000003, -0.2892}}, {{-0.10709999999999997, -0.3987}}, 
  {{-0.12880000000000003, -0.5081}}, {{-0.1906000000000001, -0.6009}}, {{-0.2831999999999999, -0.6629}}, 
  {{-0.3924000000000001, -0.6846}}, {{-0.5016, -0.6629}}, {{-0.5942000000000001, -0.6009}}, {{-0.6560999999999999, -0.5081}}, 
  {{-0.7828999999999999, -0.1023}}, {{-0.6881999999999999, -0.0074}}, {{-0.09660000000000002, -0.0074}}, 
  {{-0.0020000000000000018, -0.1023}}, {{-0.0020000000000000018, -0.6951}}, {{-0.09660000000000002, -0.79}}, 
  {{-0.6881999999999999, -0.79}}, {{-0.7828999999999999, -0.6951}}, {{-0.9021999999999999, -0.6341}}, 
  {{-0.8742000000000001, -0.3987}}, {{-0.9021999999999999, -0.1632}}, {{-1.0345, 0.2448}}, {{-0.6274, 0.1122}}, 
  {{-0.3924000000000001, 0.0841}}, {{-0.15739999999999998, 0.1122}}, {{0.24960000000000004, 0.2448}}, 
  {{0.11739999999999995, -0.1632}}, {{0.08939999999999992, -0.3987}}, {{0.11739999999999995, -0.6341}}, 
  {{0.24960000000000004, -1.0421}}, {{-0.15739999999999998, -0.9095}}, {{-0.3924000000000001, -0.8815}}, {{1.4649, -2.3987}}, 
  {{1.5067, -2.2976}}, {{1.6076, -2.2557}}, {{1.7085, -2.2976}}, {{1.7503, -2.3987}}, {{1.7085, -2.4998}}, {{1.6076, -2.5416}}, 
  {{1.5067, -2.4998}}, {{1.3222, -2.3987}}, {{1.3439, -2.2892}}, {{1.4058, -2.1965}}, {{1.4984, -2.1345}}, {{1.6076, -2.1127}}, 
  {{1.7168, -2.1345}}, {{1.8094, -2.1965}}, {{1.8712, -2.2892}}, {{1.8929, -2.3987}}, {{1.8712, -2.5080999999999998}}, 
  {{1.8094, -2.6009}}, {{1.7168, -2.6629}}, {{1.6076, -2.6846}}, {{1.4984, -2.6629}}, {{1.4058, -2.6009}}, 
  {{1.3439, -2.5080999999999998}}, {{1.2171, -2.1023}}, {{1.3118, -2.0074}}, {{1.9034, -2.0074}}, {{1.998, -2.1023}}, 
  {{1.998, -2.6951}}, {{1.9034, -2.79}}, {{1.3118, -2.79}}, {{1.2171, -2.6951}}, {{1.0978, -2.6341}}, {{1.1258, -2.3987}}, 
  {{1.0978, -2.1632}}, {{0.9655, -1.7552}}, {{1.3726, -1.8878}}, {{1.6076, -1.9159}}, {{1.8426, -1.8878}}, {{2.2496, -1.7552}}, 
  {{2.1174, -2.1632}}, {{2.0894, -2.3987}}, {{2.1174, -2.6341}}, {{2.2496, -3.0421}}, {{1.8426, -2.9095}}, {{1.6076, -2.8815}}, 
  {{1.9655, -0.042100000000000026}}, {{-0.034499999999999975, -2.0421}}}];

GraphPlot[g186,Method->None]

g=ToCombinatoricaGraph[]

RecognizeGraph[g]

FastIsomorphicQ[g186,g]

ToCommonEdges[g,Graph[{"PlanarHypotraceable",186}]]//Round

GetGraphData[{"PlanarHypotraceable",186},"File"]

PlanarQ[g186]//Timing

FindHamiltonianCycle[g186]//Timing

FastHamiltonianQ[g186]//Timing

HypohamiltonianQ[g186,False]//Timing

TraceableQ[g186,{}]//Timing

HypotraceableQ[g186]//Timing

340 Vertices (5 70-Araya-Wiener graphs)

340 Vertices (5 70-Araya-Wiener graphs)

http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p85

GraphData

GraphDataString

GraphDataString[g162,
"StandardName"->{"PlanarHypotraceable",162},
"VertexCoordinates"->None,
"Name"->"planar hypotraceable graph 162",
"Classes"->{"Hypotraceable"},
"Skip"->{"LineGraph"},
"Source"->"HypotraceableGraph",
TimeConstraint->30,
Debug->True
]

GraphDataString[g186,
"StandardName"->{"PlanarHypotraceable",186},
"VertexCoordinates"->None,
"Name"->"planar hypotraceable graph 186",
"Classes"->{"Hypotraceable"},
"Skip"->{"LineGraph"},
"Source"->"HypotraceableGraph",
TimeConstraint->30,
Debug->True
]