Dyck Graph

Author

Eric W. Weisstein
June 19, 2023

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

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Embeddings

<<MathWorld`Graphs`

Names

GraphData["DyckGraph","StandardName"]

GraphData["DyckGraph","StandardNames"]

Primary

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

Read and Wilson construction

Graph[Range[32],UndirectedEdge@@@{{1,2},{1,3},{1,11},{2,4},{2,12},{3,6},{3,9},{4,5},{4,10},{5,6},{5,21},{6,22},{7,11},{7,13},{7,16},{8,12},{8,14},{8,15},{9,13},{9,17},{10,14},{10,18},{11,27},{12,28},{13,19},{14,20},{15,17},{15,23},{16,18},{16,24},{17,26},{18,25},{19,23},{19,25},{20,24},{20,26},{21,25},{21,31},{22,26},{22,32},{23,29},{24,30},{27,28},{27,30},{28,29},{29,31},{30,32},{31,32}},VertexCoordinates->(10^(-4)){{7747,-13120},{7747,-7565},{8142,-14076},{8142,-6609},{9871,-11630},{9871,-9054},{10697,-13626},{10697,-7059},{12068,-18004},{12068,-2681},{12517,-15447},{12517,-5237},{13023,-18400},{13023,-2285},{14512,-16274},{14512,-4411},{17086,-16274},{17086,-4411},{18574,-18400},{18574,-2285},{19080,-15447},{19080,-5237},{19530,-18004},{19530,-2681},{20901,-13626},{20901,-7059},{21727,-11630},{21727,-9054},{23455,-14076},{23455,-6609},{23851,-13120},{23851,-7565}}]

RecognizeGraph@g

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["DyckGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

g0=LabelGraph@CenterGraph@RotateGraph[,Pi/2]

v=GraphEmbedding[g0];
v=Join[v,{v[[18]]-{1,0},v[[12]]+{1,0}}];
ee=Join[EdgeList[g0],UndirectedEdge@@@{{18,31},{12,32},{31,29},{30,32},{32,8},{31,4},{24,1},{28,11},{16,25},{7,19}}];
g=Graph[Range@Length@v,ee,VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

v=Join[CirclePoints[1,8],CirclePoints[3/4,8],CirclePoints[x,8],CirclePoints[1/4,8]];

Solve[RegionDistance[Line@v[[{15,13}]],v[[22]]]==0,x]

N[%]

RegionDistance[Line@v[[{14,16}]],v[[23]]]

v=Join[CirclePoints[1,8],CirclePoints[3/4,8],CirclePoints[(3/(4 (Sqrt[2]))),8],CirclePoints[1/4,8]];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[{1,8},1,9,18,11,20,13,22,15,24,9],e[16,17,10,19,12,21,14,23,16],e[32,27,30,25,28,31,26,29,32,24],{8,16},{2,10},{3,11},{4,12},{5,13},{6,14},{7,15},{17,25},{18,26},{19,27},{20,28},{21,29},{22,30},{23,31}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

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

RecognizeGraph[g]

Symmetric construction from Ed Pegg

From Ed Pegg, Jr.
Oct. 17, 2007

g=Graph[Range[32],UndirectedEdge@@@{{1,2},{1,3},{1,7},{2,4},{2,8},{3,6},{3,13},{4,5},{4,14},{5,9},{5,11},{6,10},{6,12},{7,9},{7,25},{8,10},{8,26},{9,19},{10,20},{11,18},{11,21},{12,17},{12,22},{13,15},{13,23},{14,16},{14,24},{15,16},{15,22},{16,21},{17,18},{17,19},{18,20},{19,29},{20,30},{21,27},{22,28},{23,25},{23,27},{24,26},{24,28},{25,31},{26,32},{27,30},{28,29},{29,31},{30,32},{31,32}},VertexCoordinates->(10^(-4)){{2829,-16657},{2829,-5546},{6181,-15268},{6181,-6935},{9533,-13879},{9533,-8324},{10683,-24514},{10683,2311},{12071,-21161},{12071,-1042},{12885,-12490},{12885,-9713},{13460,-17808},{13460,-4395},{14848,-14455},{14848,-7748},{17625,-14455},{17625,-7748},{19013,-17808},{19013,-4395},{19588,-12490},{19588,-9713},{20402,-21161},{20402,-1042},{21790,-24514},{21790,2311},{22940,-13879},{22940,-8324},{26292,-15268},{26292,-6935},{29644,-16657},{29644,-5546}}]

RecognizeGraph@g

g=Graph[Range[32],UndirectedEdge@@@{{1,2},{1,3},{1,5},{2,4},{2,6},{3,7},{3,23},{4,8},{4,24},{5,10},{5,12},{6,9},{6,13},{7,11},{7,16},{8,14},{8,15},{9,10},{9,29},{10,30},{11,12},{11,19},{12,17},{13,14},{13,18},{14,22},{15,17},{15,20},{16,18},{16,21},{17,26},{18,25},{19,20},{19,25},{20,27},{21,22},{21,28},{22,26},{23,24},{23,28},{24,27},{25,29},{26,30},{27,31},{28,32},{29,31},{30,32},{31,32}},VertexCoordinates->(10^(-4)){{6382,3783},{6382,7686},{7874,177},{7874,11291},{10283,3783},{10283,7686},{10633,-2582},{10633,14051},{12233,4306},{12233,7163},{14237,-4076},{14237,-173},{14237,11642},{14237,15544},{14760,1778},{14760,9690},{17616,1778},{17616,9690},{18138,-4076},{18138,-173},{18138,11642},{18138,15544},{20142,4306},{20142,7163},{21743,-2582},{21743,14051},{22093,3783},{22093,7686},{24501,177},{24501,11291},{25994,3783},{25994,7686}}]

RecognizeGraph@g

Other constructions from Ed Pegg

From Ed Pegg, Jr.
Oct. 17, 2007

Degenerate

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

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

Degenerate construction

Degenerate common

q4=GraphData["Q4","Graph",{"Labeled","UnitDistance"}][[1]]

v=GraphEmbedding[q4]

Degenerate 1

{g1=CenterGraph[LabelGraph[]],q4}

nf=Nearest[v]

v1=nf[#][[1]]&/@GraphEmbedding[g1]

g1new=Graph[Range[32],EdgeList[g1],VertexCoordinates->v1]

RecognizeGraph[g1new]

DegenerateGraphEmbeddingTypes[g1new]

Degenerate 2

{g2=CenterGraph[LabelGraph[]],q4}

v2=nf[#][[1]]&/@GraphEmbedding[g2]

g2new=Graph[Range[32],EdgeList[g2],VertexCoordinates->v2]

RecognizeGraph[g1new]

DegenerateGraphEmbeddingTypes[g1new]

gs={g1new,g2new}

Bilateral

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

Circular

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

Planar

GraphData["DyckGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["DyckGraph"],"CrossingNumber"->cn,RandomSeed->n]],{n,250}],Max[OuterFace[#]]<=GraphData["DyckGraph","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["DyckGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["DyckGraph","Circulant"]

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

Line graph

GraphData["DyckGraph","Line"]

LineGraphQ[GraphData["DyckGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["DyckGraph"]]

Halin

GraphData["DyckGraph","Halin"]

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

Rigid

GraphData["DyckGraph","Rigid"]

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

Flexible

GraphData["DyckGraph","Flexible"]

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

LCF

GraphData["DyckGraph","LCF"]

GraphData["DyckGraph","HamiltonianCycleCount"]

GraphData["DyckGraph","LCFSignature"]

GraphData["DyckGraph","LCFNotations"]

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

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

LCF compute all

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

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

lcf

LCFGraph@@@lcf//StyleGraphs

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

UnitDistance

GraphData["DyckGraph","UnitDistance"]

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

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

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

UnitDistance construction

Gerbracht

rA=Sin[1/2*(Pi-2*k/n*Pi)]/Sin[k*2*Pi/n]/.{n->8,k->1};
rB=Sin[1/2*(Pi-2*k/n*Pi)]/Sin[k*2*Pi/n]/.{n->8,k->3};
A1={rA*Sin[0*2*Pi/8],rA*Cos[0*2*Pi/8]};
A2={rA*Sin[7*2*Pi/8],rA*Cos[7*2*Pi/8]};
A3={rA*Sin[6*2*Pi/8],rA*Cos[6*2*Pi/8]};
A4={rA*Sin[5*2*Pi/8],rA*Cos[5*2*Pi/8]};
A5={rA*Sin[4*2*Pi/8],rA*Cos[4*2*Pi/8]};
A6={rA*Sin[3*2*Pi/8],rA*Cos[3*2*Pi/8]};
A7={rA*Sin[2*2*Pi/8],rA*Cos[2*2*Pi/8]};
A8={rA*Sin[1*2*Pi/8],rA*Cos[1*2*Pi/8]};

B1=FullSimplify[{-Sqrt[rB^2-(1/20)^2],1/20}];
B2={Re[(B1[[1]]+I*B1[[2]])*Exp[I*1*(2*Pi)/8]],Im[(B1[[1]]+I*B1[[2]])*Exp[I*1*(2*Pi)/8]]};B3={Re[(B1[[1]]+I*B1[[2]])*Exp[I*2*(2*Pi)/8]],Im[(B1[[1]]+I*B1[[2]])*Exp[I*2*(2*Pi)/8]]};
B4={Re[(B1[[1]]+I*B1[[2]])*Exp[I*3*(2*Pi)/8]],Im[(B1[[1]]+I*B1[[2]])*Exp[I*3*(2*Pi)/8]]};
B5={Re[(B1[[1]]+I*B1[[2]])*Exp[I*4*(2*Pi)/8]],Im[(B1[[1]]+I*B1[[2]])*Exp[I*4*(2*Pi)/8]]};
B6={Re[(B1[[1]]+I*B1[[2]])*Exp[I*5*(2*Pi)/8]],Im[(B1[[1]]+I*B1[[2]])*Exp[I*5*(2*Pi)/8]]};
B7={Re[(B1[[1]]+I*B1[[2]])*Exp[I*6*(2*Pi)/8]],Im[(B1[[1]]+I*B1[[2]])*Exp[I*6*(2*Pi)/8]]};
B8={Re[(B1[[1]]+I*B1[[2]])*Exp[I*7*(2*Pi)/8]],Im[(B1[[1]]+I*B1[[2]])*Exp[I*7*(2*Pi)/8]]};

P1=(xb1-B1[[1]])^2+(yb1-B1[[2]])^2-1;
P2=(A2[[1]]-xa2)^2+(A2[[2]]-ya2)^2-1;
P3=(xb1-xa2)^2+(yb1-ya2)^2-1;
P4=(xb1-ya2)^2+(yb1+xa2)^2-1;

eqn=Thread[{P1,P2,P3,P4}==0]

vars=Variables[Subtract@@@eqn]

Length[T=NSolve[eqn,vars,WorkingPrecision->100];]

T//N

DyckUDVertices[T_]:=Module[{},
Q3=Factor[Resultant[P3,P1,yb1]];
Q4=Factor[Resultant[P4,P1,yb1]];
R3=Factor[Resultant[Q3,P2,ya2]];
R4=Factor[Resultant[Q4,P2,ya2]];
(
a2num={xa2,ya2}/.#;
b1num={xb1,yb1}/.#;
b1=b1num;
b2={Re[(b1num[[1]]+I*b1num[[2]])*Exp[I*1*(2*Pi)/8]],Im[(b1num[[1]]+I*b1num[[2]])*Exp[I*1*(2*Pi)/8]]};
b3={Re[(b1num[[1]]+I*b1num[[2]])*Exp[I*2*(2*Pi)/8]],Im[(b1num[[1]]+I*b1num[[2]])*Exp[I*2*(2*Pi)/8]]};
b4={Re[(b1num[[1]]+I*b1num[[2]])*Exp[I*3*(2*Pi)/8]],Im[(b1num[[1]]+I*b1num[[2]])*Exp[I*3*(2*Pi)/8]]};
b5={Re[(b1num[[1]]+I*b1num[[2]])*Exp[I*4*(2*Pi)/8]],Im[(b1num[[1]]+I*b1num[[2]])*Exp[I*4*(2*Pi)/8]]};
b6={Re[(b1num[[1]]+I*b1num[[2]])*Exp[I*5*(2*Pi)/8]],Im[(b1num[[1]]+I*b1num[[2]])*Exp[I*5*(2*Pi)/8]]};
b7={Re[(b1num[[1]]+I*b1num[[2]])*Exp[I*6*(2*Pi)/8]],Im[(b1num[[1]]+I*b1num[[2]])*Exp[I*6*(2*Pi)/8]]};
b8={Re[(b1num[[1]]+I*b1num[[2]])*Exp[I*7*(2*Pi)/8]],Im[(b1num[[1]]+I*b1num[[2]])*Exp[I*7*(2*Pi)/8]]};a1={Re[(a2num[[1]]+I*a2num[[2]])*Exp[I*7*(2*Pi)/8]],Im[(a2num[[1]]+I*a2num[[2]])*Exp[I*7*(2*Pi)/8]]};
a2=a2num;
a3={Re[(a2num[[1]]+I*a2num[[2]])*Exp[I*1*(2*Pi)/8]],Im[(a2num[[1]]+I*a2num[[2]])*Exp[I*1*(2*Pi)/8]]};
a4={Re[(a2num[[1]]+I*a2num[[2]])*Exp[I*2*(2*Pi)/8]],Im[(a2num[[1]]+I*a2num[[2]])*Exp[I*2*(2*Pi)/8]]};
a5={Re[(a2num[[1]]+I*a2num[[2]])*Exp[I*3*(2*Pi)/8]],Im[(a2num[[1]]+I*a2num[[2]])*Exp[I*3*(2*Pi)/8]]};
a6={Re[(a2num[[1]]+I*a2num[[2]])*Exp[I*4*(2*Pi)/8]],Im[(a2num[[1]]+I*a2num[[2]])*Exp[I*4*(2*Pi)/8]]};
a7={Re[(a2num[[1]]+I*a2num[[2]])*Exp[I*5*(2*Pi)/8]],Im[(a2num[[1]]+I*a2num[[2]])*Exp[I*5*(2*Pi)/8]]};
a8={Re[(a2num[[1]]+I*a2num[[2]])*Exp[I*6*(2*Pi)/8]],Im[(a2num[[1]]+I*a2num[[2]])*Exp[I*6*(2*Pi)/8]]};
ToExpression[Flatten[Outer[StringJoin,{"A","B","a","b"},ToString/@Range[8]]]])&/@T
]

GraphicsGrid[Partition[Graphics[{Red,Point[#]}]&/@DyckUDVertices[T],3],Dividers->All]

gs=Graph[Range[32],UnitEdges[#],VertexCoordinates->#]&/@DyckUDVertices[T]

RecognizeGraph/@gs

UnitDistance construction exact

(T=Solve[eqn,vars,Reals])//Timing

ByteCount[T]

RootReduce[(NumericalApproximation[Root, (RowDefault["-0.0305", FEExpressionsNumericalApproximationElider]), Root[{-2+((#1)^2)&,2-4 ((#2)^2)+((#2)^4)&,79201-798 ((#3)^2)+((#3)^4)&,-475011840320445256932160+11656750008190168088300 #1-121402085409694797947668 #2+14641124454007845613544 #1 #2+134991697641608193817860 ((#2)^2)-560674212022928820960 #1 ((#2)^2)+35378695568621267291811 ((#2)^3)-4167059675000841724342 #1 ((#2)^3)-6257920008178851082800 #3+«269»+79760583265580800 ((#2)^2) ((#3)^3) ((#4)^5)+15938485195685120000 #1 ((#2)^2) ((#3)^3) ((#4)^5)+798751379073472000 ((#2)^3) ((#3)^3) ((#4)^5)+3430563308041024000 #1 ((#2)^3) ((#3)^3) ((#4)^5)-373253264310332187136000 ((#4)^6)&,«1»&},{2,4,3,1,1}], ]),Method->"Recursive"]//Timing

Length[T]

T//N

GroebnerBasis[Subtract@@@Thread[eqn],{xa2,ya2,xb1,yb1},MonomialOrder->EliminationOrder]

gs=Graph[Range[32],UnitEdges[#],VertexCoordinates->#]&/@DyckUDVertices[T]

RecognizeGraph/@gs

vertices=GraphEmbedding/@gs;

edges=EdgeList[]

1

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->48,WorkingPrecision->5000]&/@v[[i]]//Timing}],
{i,Length[v]}];

g1=Graph[Range[32],edges,VertexCoordinates->{{0,(NumericalApproximation[Root, (RowDefault["1.31", FEExpressionsNumericalApproximationElider]), Root[1-4 ((#1)^2)+2 ((#1)^4)&,4], ])},{(NumericalApproximation[Root, (RowDefault["-0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,4], ])},{(NumericalApproximation[Root, (RowDefault["-1.31", FEExpressionsNumericalApproximationElider]), Root[1-4 ((#1)^2)+2 ((#1)^4)&,1], ]),0},{(NumericalApproximation[Root, (RowDefault["-0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,1], ]),(NumericalApproximation[Root, (RowDefault["-0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,1], ])},{0,(NumericalApproximation[Root, (RowDefault["-1.31", FEExpressionsNumericalApproximationElider]), Root[1-4 ((#1)^2)+2 ((#1)^4)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,4], ]),(NumericalApproximation[Root, (RowDefault["-0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,1], ])},{(NumericalApproximation[Root, (RowDefault["1.31", FEExpressionsNumericalApproximationElider]), Root[1-4 ((#1)^2)+2 ((#1)^4)&,4], ]),0},{(NumericalApproximation[Root, (RowDefault["0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,4], ]),(NumericalApproximation[Root, (RowDefault["0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,4], ])},{(NumericalApproximation[Root, (RowDefault["-0.539", FEExpressionsNumericalApproximationElider]), Root[79201-319200 ((#1)^2)+160000 ((#1)^4)&,2], ]),(1/20)},{(NumericalApproximation[Root, (RowDefault["-0.416", FEExpressionsNumericalApproximationElider]), Root[19601-8000 #1-160000 ((#1)^2)+160000 ((#1)^4)&,2], ]),(NumericalApproximation[Root, (RowDefault["-0.346", FEExpressionsNumericalApproximationElider]), Root[19601+8000 #1-160000 ((#1)^2)+160000 ((#1)^4)&,2], ])},{-(1/20),(NumericalApproximation[Root, (RowDefault["-0.539", FEExpressionsNumericalApproximationElider]), Root[79201-319200 ((#1)^2)+160000 ((#1)^4)&,2], ])},{(NumericalApproximation[Root, (RowDefault["0.346", FEExpressionsNumericalApproximationElider]), Root[19601-8000 #1-160000 ((#1)^2)+160000 ((#1)^4)&,3], ]),(NumericalApproximation[Root, (RowDefault["-0.416", FEExpressionsNumericalApproximationElider]), Root[19601-8000 #1-160000 ((#1)^2)+160000 ((#1)^4)&,2], ])},{(NumericalApproximation[Root, (RowDefault["0.539", FEExpressionsNumericalApproximationElider]), Root[79201-319200 ((#1)^2)+160000 ((#1)^4)&,3], ]),-(1/20)},{(NumericalApproximation[Root, (RowDefault["0.416", FEExpressionsNumericalApproximationElider]), Root[19601+8000 #1-160000 ((#1)^2)+160000 ((#1)^4)&,3], ]),(NumericalApproximation[Root, (RowDefault["0.346", FEExpressionsNumericalApproximationElider]), Root[19601-8000 #1-160000 ((#1)^2)+160000 ((#1)^4)&,3], ])},{(1/20),(NumericalApproximation[Root, (RowDefault["0.539", FEExpressionsNumericalApproximationElider]), Root[79201-319200 ((#1)^2)+160000 ((#1)^4)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-0.346", FEExpressionsNumericalApproximationElider]), Root[19601+8000 #1-160000 ((#1)^2)+160000 ((#1)^4)&,2], ]),(NumericalApproximation[Root, (RowDefault["0.416", FEExpressionsNumericalApproximationElider]), Root[19601+8000 #1-160000 ((#1)^2)+160000 ((#1)^4)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-0.886", FEExpressionsNumericalApproximationElider]), Root[674436093967643389061775104596856934721-1669075572326047475233407639527896482996352 ((#1)^2)+332629101243811592577587555296842137707782400 ((#1)^4)-22405687138123591689107015490369858648953384960 ((#1)^6)+«27»+43172290548157882260341793733309989528797184000000000000 ((#1)^44)-8324694379005537859743796075912736276480000000000000000 ((#1)^46)+749776790168782612536549821740597903360000000000000000 ((#1)^48)&,7], ]),(NumericalApproximation[Root, (RowDefault["0.843", FEExpressionsNumericalApproximationElider]), Root[40147937324927520001+80293847065727840 #1-4487029870926449817664 ((#1)^2)-1092656396364759159680 ((#1)^3)+168623062157087326273664 ((#1)^4)+«21»+174283391519320073830400000 ((#1)^19)+9659016694453959466680320000 ((#1)^20)-264959149875522437120000000 ((#1)^21)-4416810605845518745600000000 ((#1)^22)+96466359185047552000000000 ((#1)^23)+865896523938503065600000000 ((#1)^24)&,18], ])},{(NumericalApproximation[Root, (RowDefault["-1.22", FEExpressionsNumericalApproximationElider]), Root[39347999779588156801+12591359929468210176320 #1-1562898906633098015974784 ((#1)^2)-407794194899431658264665600 ((#1)^3)+46943421338270029291135066880 ((#1)^4)+«64»+266162486764802259744782204464890118144000000000000000 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RecognizeGraph[g1]

UnitDistanceGraphEmbeddingQ[g1]

2

v=vertices[[2]];

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->48,WorkingPrecision->5000]&/@v[[i]]//Timing}],
{i,Length[v]}];//Timing

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RecognizeGraph[g2]

UnitDistanceGraphEmbeddingQ[g2]

3

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->48,WorkingPrecision->5000]&/@v[[i]]//Timing}],
{i,Length[v]}];//Timing

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RecognizeGraph[g3]

UnitDistanceGraphEmbeddingQ[g3]

4

v=vertices[[4]];

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->48,WorkingPrecision->5000]&/@v[[i]]//Timing}],
{i,Length[v]}];//Timing

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RecognizeGraph[g4]

UnitDistanceGraphEmbeddingQ[g4]

5

v=vertices[[5]];

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->48,WorkingPrecision->5000]&/@v[[i]]//Timing}],
{i,Length[v]}];//Timing

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RecognizeGraph[g5]

UnitDistanceGraphEmbeddingQ[g5]

6

v=vertices[[6]];

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->48,WorkingPrecision->5000]&/@v[[i]]//Timing}],
{i,Length[v]}];//Timing

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(RowDefault["1.69", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "-3"])]), Root[6004028286680321777496337890625-7713975542726877419727294921875000 #1+2206067313781737768489943359375000000 ((#1)^2)+«66»+62308355525958478955378900992000000000000000000000000000000000 ((#1)^47)+114406858851437776571128818014617600000000000000000000000000000000 ((#1)^48)&,24], ])}}]

RecognizeGraph[g6]

UnitDistanceGraphEmbeddingQ[g6]

1-6

gs=Table[Symbol["g"<>ToString[i]],{i,6}]

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

gs={RotateGraph[LabelGraph[],{14,13}],LabelGraph[],RotateGraph[LabelGraph[],{11,13}]}

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["DyckGraph","Matchstick"]

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

xyz

GraphData["DyckGraph","Graph","XYZ"]//StyleGraphs

xyz construction

Projection2D[v0_,o_:{0,0,0},r_:1]:=Module[{v=v0-o},{v[[1]],v[[2]]}r/(r-v[[3]])]

g=With[{g=XYZGraph[4]},Annotate[g,VertexCoordinates->(Projection2D[#,20{1,1,2},20]&/@GraphEmbedding[g])]]

GraphEmbedding[g]

3D default

Graph3D[GraphData["DyckGraph","Edges"]]

3D All

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

3D xyz

GraphData["DyckGraph","Graph",{"3D","XYZ"}]

3D xyz construction

From David Eppstein
http://11011110.livejournal.com/120326.html

g=XYZGraph[4]

RecognizeGraph@g

3D UnitDistance

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

3D Polyhedron

GraphData["DyckGraph","Polyhedral"]

Properties

Matrices

GraphicsRow[GraphMatrixPlot[GraphData["DyckGraph"]]]

Hyperbolic Construction

Graphics[HyperbolicTiling[8,3]]

p=Select[First/@HyperbolicTilingVertices[8,3],Norm[#]<.8&];

Graphics[MapIndexed[{{Red,PointSize[.015],Point[#]},Text[First[#2],#+.05]}&,p]]

ShowLabeledGraph[g=Graph[List/@Flatten[Join[
{
Partition[{9,10,16,18,24,23,17,15},2,1,1]},
{{{12,14},{22,26},{28,32},{27,29},{19,21},{7,11},{1,5},{4,6}}},
{{{}}}
Partition[#,2,1]&/@{{20,22,18},{30,28,24},{31,27,23},{25,21,17},{13,11,15},{3,5,9},{2,6,10},{8,12,16}}],1],List/@p]]

M[g]

GraphData

GraphClassString

GraphClassString["DyckGraph"]

GraphDataTable

TableForm[GraphDataTable["DyckGraph",x],TableDepth->2]//TraditionalForm