Heawood Graph

Author

Eric W. Weisstein
June 26, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Minimal rectilinear crossing embedding (possibly based on Exoo) from
Ed Pegg, Jr.
February 27, 2007

Asymmetric unit-distance embeddings by
E. H.-A. Gerbracht
December 2009 and
Feb 18, 2010 (resent 1/31/2020)

Horizontally-symmetric unit-distance embedding by
Ed Pegg, Jr.
communicated February 23, 2018

Configuration

GraphData["HeawoodGraph","Incidence"]

lines=GraphData["HeawoodGraph","Configurations"]

Subscript@@Reverse@Flatten@Tally[Length/@lines[[1]]]

RecognizeGraph[LeviGraph[#]]&/@lines

GraphData["HeawoodConfigurationGraph","#"]

https://en.wikipedia.org/wiki/Fano_plane

These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements.

Embeddings

<<MathWorld`Graphs`

Names

GraphData["HeawoodGraph","StandardName"]

GraphData["HeawoodGraph","StandardNames"]

Primary

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

Read and Wilson

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["HeawoodGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

CVT construction

g=CubicVertexTransitiveGraph[14]

Degenerate

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

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

Incidence

GraphData["HeawoodGraph","Incidence"]

Select[GraphData[All],GraphData[#,"LeviGraph","Name"]==="HeawoodGraph"&]

lines=GraphData["HeawoodGraph","IncidenceLines"]

ConfigurationNotation[lines]

levi=LeviGraph[lines]

RecognizeGraph@levi

HoneycombToroidal construction

GraphData["HeawoodGraph","NotationRules","HoneycombToroidal"]

HoneycombToroidalGraph[1,14,5]

RecognizeGraph[%]

GeneralizedHoneycombGraph construction

With[{n=14},Cases[Table[{k,g=GeneralizedHoneycombGraph[n,k],RecognizeGraph@g},{k,n-1}],{__,"HeawoodGraph"}]]//Timing

g=GeneralizedHoneycombGraph[14,3]

RecognizeGraph@g

Bilateral

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

Circular

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

Planar

GraphData["HeawoodGraph","Planar"]

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

MinimalLocalCrossing construction

https://en.wikipedia.org/wiki/1-planar_graph

g=EdgeAdd[IndexGraph@VertexDelete[RotateGraph[SpokeGraph[3,5,VertexLabels->Automatic],Pi],{13,15}],UndirectedEdge@@@{{10,13},{7,5},{4,9},{1,11},{12,2},{13,12},{6,8},{3,10}}]

RecognizeGraph@g

LocalCrossingCount[g]//Timing

MinimalLocalCrossing

GraphData["HeawoodGraph","LocalCrossingNumber"]

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["HeawoodGraph"],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData["HeawoodGraph","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["HeawoodGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,500}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["HeawoodGraph","Circulant"]

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

Line graph

GraphData["HeawoodGraph","Line"]

LineGraphQ[GraphData["HeawoodGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["HeawoodGraph"]]

Halin

GraphData["HeawoodGraph","Halin"]

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

Rigid

GraphData["HeawoodGraph","Rigid"]

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

Flexible

GraphData["HeawoodGraph","Flexible"]

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

LCF

GraphData["HeawoodGraph","LCF"]

GraphData["HeawoodGraph","HamiltonianCycleCount"]

GraphData["HeawoodGraph","LCFSignature"]

GraphData["HeawoodGraph","LCFNotations"]

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

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

LCF compute all

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

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

Matchstick

GraphData["HeawoodGraph","Matchstick"]

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

3D default

Graph3D[GraphData["HeawoodGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["HeawoodGraph","Polyhedral"]

Properties

CrossingNumber

GraphData["HeawoodGraph","CrossingNumber"]

Dear Eric Weisstein,

we have successfully computed the crossing number of your graph (Heawood graph).
The graph has a crossing number of 3.

Please visit this link for detailed results: http://crossings.uos.de/job/gjm67RIK_EHpgnFuJA_l8w

Construction

MinimalCrossing from Exoo

Show[ImageCrop@ImageTake[Import["http://isu.indstate.edu/ge/COMBIN/RECTILINEAR/heawood.gif"],{60,550},{40,520}],ImageSize->Small]

distanceregular.org

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/heawood.am"]

RecognizeGraph[g]

VertexCount[g]

GraphDiameter[g]//Timing

RegularParameters[g]//Timing

DistanceRegularGraphQ[g]//Timing

IntersectionArray[g]//Timing

IntegerSpectrum[g]//SpectrumForm

GroupOrder[GraphAutomorphismGroup[g]]

HamiltonianGraphQ[g]

UnitDistance random

GraphData["HeawoodGraph",{"TriangleCount","SquareCount"}]

Select[FindUnitDistanceEmbedding[GraphData["HeawoodGraph"],20],UnitDistanceGraphEmbeddingQ]

g=RotateGraph[LabelGraph[],{9,8}]

UnitDistanceGraphEmbeddingQ[g]

Select[FindUnitDistanceEmbedding[GraphData["HeawoodGraph"],20],UnitDistanceGraphEmbeddingQ]

UnitDistance Pegg horizontally symmetric

http://community.wolfram.com/groups/-/m/t/518461

ca. 2016

v={{-1/2,-(1/2) (-1+Sqrt[3-4 x-4 x^2])},{x,1/2},{x,-1/2},{-1/2,1/2 (-1+Sqrt[3-4 x-4 x^2])},{1/2,1/2 (-1+Sqrt[3-4 x-4 x^2])},{-x,-1/2},{-x,1/2},{1/2,-(1/2) (-1+Sqrt[3-4 x-4 x^2])},{1/2 (-1+Sqrt[3-4 x (1+x)]) Sqrt[(7+8 Sqrt[3-4 x (1+x)]-8 x (-1+x (1+2 x (2+x))))/(13+8 x (1+x) (-3+2 x (1+x)))],-(1/2) Sqrt[(7+8 Sqrt[3-4 x (1+x)]-8 x (-1+x (1+2 x (2+x))))/(13+8 x (1+x) (-3+2 x (1+x)))]},{Sqrt[3-4 x^2]/(2 Sqrt[1+4 x^2]),-((x Sqrt[3-4 x^2])/Sqrt[1+4 x^2])},{-Sqrt[3-4 x^2]/(2 Sqrt[1+4 x^2]),-((x Sqrt[3-4 x^2])/Sqrt[1+4 x^2])},{-(1/2) (-1+Sqrt[3-4 x (1+x)]) Sqrt[(7+8 Sqrt[3-4 x (1+x)]-8 x (-1+x (1+2 x (2+x))))/(13+8 x (1+x) (-3+2 x (1+x)))],-(1/2) Sqrt[(7+8 Sqrt[3-4 x (1+x)]-8 x (-1+x (1+2 x (2+x))))/(13+8 x (1+x) (-3+2 x (1+x)))]},{1/2,-((x Sqrt[3-4 x^2])/Sqrt[1+4 x^2])-Sqrt[(8 x^2+Sqrt[3+8 x^2-16 x^4])/(2+8 x^2)]},{-1/2,-((x Sqrt[3-4 x^2])/Sqrt[1+4 x^2])-Sqrt[(8 x^2+Sqrt[3+8 x^2-16 x^4])/(2+8 x^2)]}}/.x->Root[-153027+1353114 #1-1986024 #1^2+2763312 #1^3+10497808 #1^4+31063072 #1^5-8457728 #1^6-40590336 #1^7+44468736 #1^8+164975616 #1^9+119705600 #1^10-494067712 #1^11-763912192 #1^12+414924800 #1^13+1154613248 #1^14+48496640 #1^15-864485376 #1^16-376307712 #1^17+253231104 #1^18+271581184 #1^19+89128960 #1^20+10485760 #1^21&,1];

g=GraphData[gn="HeawoodGraph"];

g1=System`Graph[VertexList[g],EdgeList[g],VertexCoordinates->v[[{13,12,4,5,9,14,11,3,2,1,8,7,6,10}]]]

GraphEmbedding[g1][[1]]

GraphEmbedding[g1][[1]]//RootReduce

RecognizeGraph[g1]

dist=EuclideanDistance@@@(List@@@EdgeList[g1]/.i_Integer:>GraphEmbedding[g1][[i]]);

Do[Print[{i,RootReduce[dist[[i]]]//Timing}],{i,Length[dist]}]

Block[{$MaxExtraPrecision=100},N[dist-1,10^6]]//Union

Derived from scratch by Eric Weisstein, Oct. 16, 2020

(soln=Solve[{
(1/2+x)^2+(b-1)^2==1,
(x-e)^2+(f-1)^2==1,
(e+x)^2+f^2==1,
(e-1/2)^2+(a-f)^2==1,
(1/2+c)^2+(a-d)^2==1,
(1/2+c)^2+(b-d)^2==1,
(1/2+x)^2+h^2==1,
(c-1/2)^2+(h-d)^2==1},{x,h,a,b,c,d,e,f},Reals])//Timing

gs=Select[Graph[Range[14],UndirectedEdge@@@{{1,5},{1,9},{1,7},{2,6},{2,10},{2,8},{3,4},{3,10},{3,11},{4,9},{4,12},{5,10},{5,13},{6,7},{6,12},{7,11},{8,9},{8,14},{11,14},{12,13},{13,14}},VertexLabels->"Name",VertexCoordinates->({{-x,1},{x,1},{-1/2,h},{1/2,h},{-e,f},{-1/2,b},{1/2,b},{e,f},{-x,0},{x,0},{-c,d},{c,d},{-1/2,a},{1/2,a}}/.#)]&/@soln,DuplicateFreeQ[Floor[Round[10^4GraphEmbedding[#]]]]&]

g=;

RecognizeGraph/@gs

UnitDistanceGraphEmbeddingQ[g]

emb=GraphEmbedding[gs[[1]]]

ByteCount[emb]

RootReduce/@emb

ByteCount[%]

N[EdgeLengths[g]-1,1000]

EdgeLengths[g]^2

RootReduce[EdgeLengths[g]^2]//Timing

UnitDistance Gerbracht

https://arxiv.org/abs/0912.5395

These all have 6 fixed points:

P2={0,2};
P5={0,0};
P7={1,1};
l3={0,1};
l5={1,0};
l7={1,2};

gs={g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11}

Equations

(eqns={
(xl4-1)^2+yl4^2-4==0,
xP4-1/2 (xl4+1)==0,
yP4-1/2yl4==0,
xP3^2+(yP3-1)^2-1==0,
(xP3-xl4)^2+(yP3-yl4)^2-1==0,
(xP6-1)^2+(yP6-2)^2-1==0,
(xP6-xl4)^2+(yP6-yl4)^2-1==0,
xl2^2+(yl2-2)^2-1==0,
(xl2-xP4)^2+(yl2-yP4)^2-1==0,
(xl1-1)^2+(yl1-1)^2-1==0,
(xl1-xP3)^2+(yl1-yP3)^2-1==0,
xl6^2+yl6^2-1==0,
(xl6-xP6)^2+(yl6-yP6)^2-1==0,
(xP1-xl2)^2+(yP1-yl2)^2-1==0,
(xP1-xl6)^2+(yP1-yl6)^2-1==0,
(xP1-xl1)^2+(yP1-yl1)^2-1==0
})//Column//TraditionalForm

Length[eqns]

vars=Union[Flatten[Variables/@eqns[[All,1]]]]

Length[vars]

Solve

(solns=Solve[eqns,vars,Reals])//Timing

NSolve

(nsoln=NSolve[eqns,vars,Reals])//Timing

Length[nsoln]

Length[Union[nsoln]]

(DeleteDuplicatesBy)

Length[nsoln2=Union[nsoln,SameTest->Function[SameQ@@({##}/.x_?InexactNumberQ:>Round[10^4x])]]]

good=Select[ToHeadwoodGraph[#,fixed]&/@nsoln2,UnitDistanceGraphEmbeddingQ]

Length[good]

GrobnerBasis

(gb=GroebnerBasis[eqns,vars,MonomialOrder->EliminationOrder])//Timing

ToHeadwoodGraph

edges=UndirectedEdge@@@{{1,8},{1,9},{1,13},{2,14},{2,9},{2,10},{3,8},{3,10},{3,11},{4,11},{4,12},{4,9},{5,12},{5,13},{5,10},{6,13},{6,14},{6,11},{7,8},{7,14},{7,12}};

ToHeadwoodGraph[v_List]:=Graph[Range[14],edges,VertexCoordinates->v,VertexLabels->"Name"]

fixed=Flatten[{
Thread[{xP2,yP2}->{0,2}],
Thread[{xP5,yP5}->{0,0}],
Thread[{xP7,yP7}->{1,1}],
Thread[{xl3,yl3}->{0,1}],
Thread[{xl5,yl5}->{1,0}],
Thread[{xl7,yl7}->{1,2}]}];

ToHeadwoodGraph[soln_List,fixed_List]:=ToHeadwoodGraph[{{xP1,yP1},{xP2,yP2},{xP3,yP3},{xP4,yP4},{xP5,yP5},{xP6,yP6},{xP7,yP7},{xl1,yl1},{xl2,yl2},{xl3,yl3},{xl4,yl4},{xl5,yl5},{xl6,yl6},{xl7,yl7}}/.soln/.fixed]

Gerbracht 1

soln=FindRoot[
eqns,
{
{xP1,0.400182002388641},{yP1,1.043762692468723},
{xP3,-0.369307668700666},{yP3,0.070692814060453},
{xP4,0.134937917545110},{yP4,0.501664821866961},
{xP6,0.106134457655163},{yP6,1.551664866189844},
{xl1,0.630692331299334},{yl1,0.070692814060453},
{xl2,-0.588810425254328},{yl2,1.191728830706045},
{xl4,-0.730124164909779},{yl4,1.003329643733922},
{xl6,-0.574170534719569},{yl6,0.818735730904572}
},
WorkingPrecision->25000
];//Timing

g1=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g1]

UnitDistanceGraphEmbeddingQ[g1]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->7500]&/@v[[i]]//Timing}],
{i,Length[v]}];

g1=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["0.400", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,9], ]),(NumericalApproximation[Root, (RowDefault["1.04", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,6], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["-0.369", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,6], ]),(NumericalApproximation[Root, (RowDefault["0.0707", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.135", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.502", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,1], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.106", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,3], ]),(NumericalApproximation[Root, (RowDefault["1.55", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,9], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["0.631", FEExpressionsNumericalApproximationElider]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,6], ]),(NumericalApproximation[Root, (RowDefault["0.0707", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-0.589", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,3], ]),(NumericalApproximation[Root, (RowDefault["1.19", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,3], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.730", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,1], ]),(NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,1], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["-0.574", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,3], ]),(NumericalApproximation[Root, (RowDefault["0.819", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,6], ])},{1,2}}]

RecognizeGraph[g1]

Max[N[Abs[EdgeLengths[g1]],10^5]-1]//Timing

Gerbracht 2

soln=FindRoot[
eqns,
{
{xP1,0.385844376323838},{yP1,0.971300460792625},
{xP3,-0.351496569091080},{yP3,1.936189169942272},
{xP4,0.136658400253077},{yP4,0.504619938316376},
{xP6,0.184497979349687},{yP6,1.421245773825144},
{xl1,0.648503430908920},{yl1,1.936189169942272},
{xl2,-0.589452825277335},{yl2,1.192197198090668},
{xl4,-0.726683199493847},{yl4,1.009239876632752},
{xl6,-0.599446494990155},{yl6,0.800414829725199}
},
WorkingPrecision->25000
];//Timing

g2=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g2]

UnitDistanceGraphEmbeddingQ[g2]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->20000]&/@v[[i]]//Timing}],
{i,Length[v]}];

g2=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["0.386", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,7], ]),(NumericalApproximation[Root, (RowDefault["0.971", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,5], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["-0.351", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,7], ]),(NumericalApproximation[Root, (RowDefault["1.94", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,8], ])},{(NumericalApproximation[Root, (RowDefault["0.137", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,2], ]),(NumericalApproximation[Root, (RowDefault["0.505", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,2], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.184", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,4], ]),(NumericalApproximation[Root, (RowDefault["1.42", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,8], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["0.649", FEExpressionsNumericalApproximationElider]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,7], ]),(NumericalApproximation[Root, (RowDefault["1.94", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,8], ])},{(NumericalApproximation[Root, (RowDefault["-0.589", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,2], ]),(NumericalApproximation[Root, (RowDefault["1.19", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,4], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.727", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,2], ]),(NumericalApproximation[Root, (RowDefault["1.01", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,2], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["-0.599", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.800", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,4], ])},{1,2}}]

Gerbracht 3

soln=FindRoot[
eqns,
{
{xP1,-0.111888421172288},{yP1,0.807281254230185},
{xP3,-0.414946144051627},{yP3,0.090154025377544},
{xP4,0.148144628769197},{yP4,0.523777077109366},
{xP6,0.254555103623971},{yP6,1.333432744228363},
{xl1,0.585053855948373},{yl1,0.090154025377544},
{xl2,0.741321136264131},{yl2,1.328849515437814},
{xl4,-0.703710742461606},{yl4,1.047554154218733},
{xl6,0.848614578463299},{yl6,0.529011622953180}
},
WorkingPrecision->25000
];//Timing

g3=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g3]

UnitDistanceGraphEmbeddingQ[g3]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->7500]&/@v[[i]]//Timing}],
{i,Length[v]}];

g3=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["-0.112", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,3], ]),(NumericalApproximation[Root, (RowDefault["0.807", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,3], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["-0.415", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,5], ]),(NumericalApproximation[Root, (RowDefault["0.0902", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,2], ])},{(NumericalApproximation[Root, (RowDefault["0.148", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,3], ]),(NumericalApproximation[Root, (RowDefault["0.524", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,3], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.255", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,5], ]),(NumericalApproximation[Root, (RowDefault["1.33", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,7], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["0.585", FEExpressionsNumericalApproximationElider]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,5], ]),(NumericalApproximation[Root, (RowDefault["0.0902", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,2], ])},{(NumericalApproximation[Root, (RowDefault["0.741", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,6], ]),(NumericalApproximation[Root, (RowDefault["1.33", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,6], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.704", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,3], ]),(NumericalApproximation[Root, (RowDefault["1.05", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,3], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["0.849", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,9], ]),(NumericalApproximation[Root, (RowDefault["0.529", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,3], ])},{1,2}}]

RecognizeGraph[g]

UnitDistanceGraphEmbeddingQ[g]

Gerbracht 4

soln=FindRoot[
eqns,
{
{xP1,1.421904368600474},{yP1,0.598683521299139},
{xP3,-0.448398217164224},{yP3,0.106166101088158},
{xP4,0.157643970813835},{yP4,0.538921441486342},
{xP6,0.023681878654164},{yP6,1.783660161015737},
{xl1,0.551601782835776},{yl1,0.106166101088158},
{xl2,0.753214201921679},{yl2,1.342224684239756},
{xl4,-0.684712058372330},{yl4,1.077842882972684},
{xl6,0.464016631427877},{yl6,0.885826487388092}
},
WorkingPrecision->10000
];//Timing

g4=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g4]

UnitDistanceGraphEmbeddingQ[g4]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->7500]&/@v[[i]]//Timing}],
{i,Length[v]}];

g4=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["1.42", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,10], ]),(NumericalApproximation[Root, (RowDefault["0.599", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,2], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["-0.448", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,4], ]),(NumericalApproximation[Root, (RowDefault["0.106", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,3], ])},{(NumericalApproximation[Root, (RowDefault["0.158", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,4], ]),(NumericalApproximation[Root, (RowDefault["0.539", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,4], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.0237", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,2], ]),(NumericalApproximation[Root, (RowDefault["1.78", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,10], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["0.552", FEExpressionsNumericalApproximationElider]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,4], ]),(NumericalApproximation[Root, (RowDefault["0.106", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,3], ])},{(NumericalApproximation[Root, (RowDefault["0.753", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,7], ]),(NumericalApproximation[Root, (RowDefault["1.34", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,7], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.685", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,4], ]),(NumericalApproximation[Root, (RowDefault["1.08", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,4], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["0.464", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,8], ]),(NumericalApproximation[Root, (RowDefault["0.886", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,8], ])},{1,2}}]

RecognizeGraph[g4]

Gerbracht 5

soln=FindRoot[
eqns,
{
{xP1,-0.164629969634109},{yP1,1.728731712593544},
{xP3,-0.196826391803828},{yP3,1.980438356802449},
{xP4,0.164749301040245},{yP4,0.549869320736518},
{xP6,0.314574903717282},{yP6,1.271856856527629},
{xl1,0.803173608196172},{yl1,1.980438356802449},
{xl2,0.761740164689577},{yl2,1.352117355149333},
{xl4,-0.670501397919511},{yl4,1.099738641473037},
{xl6,-0.576161224535975},{yl6,0.817336065117162}
},
WorkingPrecision->10000
];//Timing

g5=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g5]

UnitDistanceGraphEmbeddingQ[g5]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->7500]&/@v[[i]]//Timing}],
{i,Length[v]}];

g5=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["-0.165", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,1], ]),(NumericalApproximation[Root, (RowDefault["1.73", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,9], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["-0.197", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,8], ]),(NumericalApproximation[Root, (RowDefault["1.98", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,9], ])},{(NumericalApproximation[Root, (RowDefault["0.165", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,5], ]),(NumericalApproximation[Root, (RowDefault["0.550", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,5], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.315", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,6], ]),(NumericalApproximation[Root, (RowDefault["1.27", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,6], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["0.803", FEExpressionsNumericalApproximationElider]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,8], ]),(NumericalApproximation[Root, (RowDefault["1.98", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,9], ])},{(NumericalApproximation[Root, (RowDefault["0.762", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,8], ]),(NumericalApproximation[Root, (RowDefault["1.35", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,8], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.671", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,5], ]),(NumericalApproximation[Root, (RowDefault["1.10", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,5], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["-0.576", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,2], ]),(NumericalApproximation[Root, (RowDefault["0.817", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,5], ])},{1,2}}]

Gerbracht 6

soln=FindRoot[
eqns,
{
{xP1,-0.162159171174261},{yP1,1.733808025238098},
{xP3,-0.193231727879694},{yP3,1.981153147750456},
{xP4,0.165394595983793},{yP4,0.550848272745721},
{xP6,0.014270178593230},{yP6,1.831664860503289},
{xl1,0.806768272120306},{yl1,1.981153147750456},
{xl2,0.762499622579621},{yl2,1.353011340465742},
{xl4,-0.669210808032414},{yl4,1.101696545491441},
{xl6,0.408620165537735},{yl6,0.912704530675680}
},
WorkingPrecision->10000
];//Timing

g6=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g6]

UnitDistanceGraphEmbeddingQ[g6]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->7500]&/@v[[i]]//Timing}],
{i,Length[v]}];

g6=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["-0.162", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,2], ]),(NumericalApproximation[Root, (RowDefault["1.73", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,10], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["-0.193", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,9], ]),(NumericalApproximation[Root, (RowDefault["1.98", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,10], ])},{(NumericalApproximation[Root, (RowDefault["0.165", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,6], ]),(NumericalApproximation[Root, (RowDefault["0.551", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,6], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.0143", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,1], ]),(NumericalApproximation[Root, (RowDefault["1.83", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,11], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["0.807", FEExpressionsNumericalApproximationElider]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,9], ]),(NumericalApproximation[Root, (RowDefault["1.98", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,10], ])},{(NumericalApproximation[Root, (RowDefault["0.762", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,9], ]),(NumericalApproximation[Root, (RowDefault["1.35", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,9], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.669", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,6], ]),(NumericalApproximation[Root, (RowDefault["1.10", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,6], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["0.409", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,7], ]),(NumericalApproximation[Root, (RowDefault["0.913", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,10], ])},{1,2}}]

Gerbracht 7

soln=FindRoot[
eqns,
{
{xP1,0.39788958050138643},{yP1,1.120875683482126},
{xP3,-0.11373768509513692},{yP3,1.993510814731878},
{xP4,0.17962533307815014},{yP4,0.571826377384660},
{xP6,0.35499397395823023},{yP6,1.235822516446733},
{xl1,0.88626231490486308},{yl1,1.993510814731878},
{xl2,-0.59903243717863292},{yl2,1.199275241292098},
{xl4,-0.64074933384369972},{yl4,1.143652754769321},
{xl6,-0.55868296106963737},{yl6,0.829381304955967}
},
WorkingPrecision->10000
];//Timing

g7=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g7]

UnitDistanceGraphEmbeddingQ[g7]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->7500]&/@v[[i]]//Timing}],
{i,Length[v]}];

g7=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["0.398", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,8], ]),(NumericalApproximation[Root, (RowDefault["1.12", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,7], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["-0.114", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,10], ]),(NumericalApproximation[Root, (RowDefault["1.99", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,11], ])},{(NumericalApproximation[Root, (RowDefault["0.180", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,7], ]),(NumericalApproximation[Root, (RowDefault["0.572", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,7], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.355", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,7], ]),(NumericalApproximation[Root, (RowDefault["1.24", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,5], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["0.886", FEExpressionsNumericalApproximationElider]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,10], ]),(NumericalApproximation[Root, (RowDefault["1.99", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,11], ])},{(NumericalApproximation[Root, (RowDefault["-0.599", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,1], ]),(NumericalApproximation[Root, (RowDefault["1.20", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,5], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.641", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,7], ]),(NumericalApproximation[Root, (RowDefault["1.14", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,7], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["-0.559", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,4], ]),(NumericalApproximation[Root, (RowDefault["0.829", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,7], ])},{1,2}}]

Gerbracht 8

soln=FindRoot[
eqns,
{
{xP1,0.374047478003772},{yP1,1.468744576326795},
{xP3,-0.864667172852078},{yP3,0.497654819678744},
{xP4,0.284999030498420},{yP4,0.699123460922175},
{xP6,0.529767447111884},{yP6,1.117457453601060},
{xl1,0.135332827147922},{yl1,0.497654819678744},
{xl2,-0.586287978493963},{yl2,1.189897286590483},
{xl4,-0.430001939003160},{yl4,1.398246921844351},
{xl6,-0.445265782381637},{yl6,0.895398449317436}
},
WorkingPrecision->10000
];//Timing

g8=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g8]

UnitDistanceGraphEmbeddingQ[g8]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->7500]&/@v[[i]]//Timing}],
{i,Length[v]}];

g8=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["0.374", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,6], ]),(NumericalApproximation[Root, (RowDefault["1.47", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,8], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["-0.865", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,3], ]),(NumericalApproximation[Root, (RowDefault["0.498", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,4], ])},{(NumericalApproximation[Root, (RowDefault["0.285", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,8], ]),(NumericalApproximation[Root, (RowDefault["0.699", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,8], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.530", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,8], ]),(NumericalApproximation[Root, (RowDefault["1.12", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,4], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["0.135", FEExpressionsNumericalApproximationElider]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,3], ]),(NumericalApproximation[Root, (RowDefault["0.498", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,4], ])},{(NumericalApproximation[Root, (RowDefault["-0.586", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,4], ]),(NumericalApproximation[Root, (RowDefault["1.19", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,2], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.430", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,8], ]),(NumericalApproximation[Root, (RowDefault["1.40", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,8], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["-0.445", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,5], ]),(NumericalApproximation[Root, (RowDefault["0.895", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,9], ])},{1,2}}]

Gerbracht 9

soln=FindRoot[
eqns,
{
{xP1,1.683478977241937},{yP1,0.926313519981179},
{xP3,0.442398409684554},{yP3,1.896818625599149},
{xP4,0.286751573157956},{yP4,0.700911322217696},
{xP6,0.531884805746619},{yP6,1.116332548454523},
{xl1,1.442398409684554},{yl1,1.896818625599149},
{xl2,0.872507318643794},{yl2,1.511398957306269},
{xl4,-0.426496853684088},{yl4,1.401822644435391},
{xl6,0.975476510630742},{yl6,0.220103560214309}
},
WorkingPrecision->10000
];//Timing

g9=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g9]

UnitDistanceGraphEmbeddingQ[g9]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->7500]&/@v[[i]]//Timing}],
{i,Length[v]}];

g9=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["1.68", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,11], ]),(NumericalApproximation[Root, (RowDefault["0.926", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,4], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["0.442", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,11], ]),(NumericalApproximation[Root, (RowDefault["1.90", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,7], ])},{(NumericalApproximation[Root, (RowDefault["0.287", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,9], ]),(NumericalApproximation[Root, (RowDefault["0.701", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,9], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.532", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,9], ]),(NumericalApproximation[Root, (RowDefault["1.12", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,3], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["1.44", FEExpressionsNumericalApproximationElider]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,11], ]),(NumericalApproximation[Root, (RowDefault["1.90", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,7], ])},{(NumericalApproximation[Root, (RowDefault["0.873", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,10], ]),(NumericalApproximation[Root, (RowDefault["1.51", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,10], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.426", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,9], ]),(NumericalApproximation[Root, (RowDefault["1.40", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,9], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["0.975", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,10], ]),(NumericalApproximation[Root, (RowDefault["0.220", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,2], ])},{1,2}}]

Gerbracht 10

soln=FindRoot[
eqns,
{
{xP1,-0.043329551699449},{yP1,1.874285721361077},
{xP3,-0.992231159457987},{yP3,0.875592097515236},
{xP4,0.370915641186566},{yP4,0.777337037260087},
{xP6,0.619416519241032},{yP6,1.075253432461959},
{xl1,0.007768840542013},{yl1,0.875592097515236},
{xl2,0.921663138782111},{yl2,1.612008945192925},
{xl4,-0.258168717626869},{yl4,1.554674074520175},
{xl6,-0.369844480601252},{yl6,0.929093676745671}
},
WorkingPrecision->10000
];//Timing

g10=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g10]

UnitDistanceGraphEmbeddingQ[g10]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->7500]&/@v[[i]]//Timing}],
{i,Length[v]}];

g10=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["-0.0433", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,4], ]),(NumericalApproximation[Root, (RowDefault["1.87", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,11], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["-0.992", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.876", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,5], ])},{(NumericalApproximation[Root, (RowDefault["0.371", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,10], ]),(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,10], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.619", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,10], ]),(NumericalApproximation[Root, (RowDefault["1.08", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,2], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["7.77", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "-3"])]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.876", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,5], ])},{(NumericalApproximation[Root, (RowDefault["0.922", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,11], ]),(NumericalApproximation[Root, (RowDefault["1.61", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,11], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.258", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,10], ]),(NumericalApproximation[Root, (RowDefault["1.55", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,10], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["-0.370", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,6], ]),(NumericalApproximation[Root, (RowDefault["0.929", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,11], ])},{1,2}}]

Gerbracht 11

soln=FindRoot[
eqns,
{
{xP1,0.010754855391715},{yP1,0.263588750091565},
{xP3,-0.964717307331239},{yP3,1.263287897434660},
{xP4,0.468775634039814},{yP4,0.847231180381246},
{xP6,0.699093018450262},{yP6,1.046346505037272},
{xl1,0.035282692668761},{yl1,1.263287897434660},
{xl2,-0.490788504254845},{yl2,1.128721259245187},
{xl4,-0.062448731920371},{yl4,1.694462360762491},
{xl6,0.995815833199486},{yl6,0.091382855882344}
},
WorkingPrecision->10000
];//Timing

g11=ToHeadwoodGraph[soln,fixed]

RecognizeGraph[g11]

UnitDistanceGraphEmbeddingQ[g11]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->80,WorkingPrecision->7500]&/@v[[i]]//Timing}],
{i,Length[v]}];

g11=Graph[Range[14],edges,VertexCoordinates->{{(NumericalApproximation[Root, (RowDefault["0.0108", FEExpressionsNumericalApproximationElider]), Root[-8078747539252971617752219013213061120+531799549331292726893402465598072422400 #1+20309822355373069692803719911203165700096 ((#1)^2)+«111»+162472784209931942779765143273523200000000 ((#1)^77)-8054864054147379177323630729625600000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,5], ]),(NumericalApproximation[Root, (RowDefault["0.264", FEExpressionsNumericalApproximationElider]), Root[-517683138906875137046229378396107833344+29109998461060116233185875305401310773248 #1-794895797731647937868026846689501536845824 ((#1)^2)+«111»+484379992161247196915945049283983360000000 ((#1)^77)-14052844370249881545605899788288000000000 ((#1)^78)+198338193685552500021004812288000000000 ((#1)^79)&,1], ])},{0,2},{(NumericalApproximation[Root, (RowDefault["-0.965", FEExpressionsNumericalApproximationElider]), Root[-28957154812074267770880000-819893521558275846517555200 #1-8555227720359662560739328000 ((#1)^2)-32938974115731518459939389440 ((#1)^3)+62707004340987958796023234560 ((#1)^4)+873835817446549434479872573440 ((#1)^5)+«104»+64623673806757588547665920000 ((#1)^76)+56127298704414755468083200000 ((#1)^77)+13371229274304065568768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,2], ]),(NumericalApproximation[Root, (RowDefault["1.26", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,6], ])},{(NumericalApproximation[Root, (RowDefault["0.469", FEExpressionsNumericalApproximationElider]), Root[-26873856+5527056384 #1-591484612608 ((#1)^2)+41660234274816 ((#1)^3)-2101276422309888 ((#1)^4)+79397142832991232 ((#1)^5)-2324393666841502080 ((#1)^6)+«102»+112033600366139200931160214986699004968960 ((#1)^75)-12402107692115498323525909880263566950400 ((#1)^76)+1007734028037881972823774118136512512000 ((#1)^77)-53410927628293472549188002140651520000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,11], ]),(NumericalApproximation[Root, (RowDefault["0.847", FEExpressionsNumericalApproximationElider]), Root[-1354036810940178563755671552+55772540958259392266870194176 #1-1051673240591974800644001398784 ((#1)^2)+11823224275996699000781033938944 ((#1)^3)-85625037234716035362852555387648 ((#1)^4)+«104»+38361698231709898790883799312017811046400 ((#1)^75)-5845142115219556636016227649920696320000 ((#1)^76)+628699923780595957661586587721400320000 ((#1)^77)-42736409295140433614127464919859200000 ((#1)^78)+1384484435958536444616153602457600000 ((#1)^79)&,11], ])},{0,0},{(NumericalApproximation[Root, (RowDefault["0.699", FEExpressionsNumericalApproximationElider]), Root[-155584800000000+28527537391163136 #1-2089585511968475232 ((#1)^2)+85290937324576471956 ((#1)^3)-2321183659377153056060 ((#1)^4)+46843012973007461873621 ((#1)^5)-750452515025622783666400 ((#1)^6)+«99»+41970852614560768994338247016448 ((#1)^73)-5859020084488659214505261137920 ((#1)^74)+688602148468272264029285646336 ((#1)^75)-65462109867525755677704192000 ((#1)^76)+4721699810034698471394508800 ((#1)^77)-229751113879682600140800000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,11], ]),(NumericalApproximation[Root, (RowDefault["1.05", FEExpressionsNumericalApproximationElider]), Root[-621057148277110010110702335937500000+27252936760252554206737403941406250000 #1-582761458107504666448427522125968750000 ((#1)^2)+8081434143517374865247260201423790625000 ((#1)^3)-81578460729927826390061616558538159968750 ((#1)^4)+«105»+31327439211035470637913977389056 ((#1)^75)-1200863487952740716716039864320 ((#1)^76)+33866724603324381796918886400 ((#1)^77)-624762668391955030867968000 ((#1)^78)+5655700632420231413760000 ((#1)^79)&,1], ])},{1,1},{(NumericalApproximation[Root, (RowDefault["0.0353", FEExpressionsNumericalApproximationElider]), Root[-6324839492301619200+1223070872696065622016 #1-71374309363529696673792 ((#1)^2)+2861071845067300018323456 ((#1)^3)-67695759631206816914866176 ((#1)^4)+«104»+959434978800413316743131299840000 ((#1)^75)-59773989760358685255118356480000 ((#1)^76)+2740597893672098247947059200000 ((#1)^77)-82203808632449796145152000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,2], ]),(NumericalApproximation[Root, (RowDefault["1.26", FEExpressionsNumericalApproximationElider]), Root[-7387029288794456064+702383368209539530752 #1-35425973795330847670272 ((#1)^2)+1130111271918785028685824 ((#1)^3)-23944245023382568213413888 ((#1)^4)+«105»+808410194661539538008942837760000 ((#1)^75)-53000501965267349376168099840000 ((#1)^76)+2542794422653504787683737600000 ((#1)^77)-79383207595912376352768000000 ((#1)^78)+1209810606414605844480000000 ((#1)^79)&,6], ])},{(NumericalApproximation[Root, (RowDefault["-0.491", FEExpressionsNumericalApproximationElider]), Root[2625245309257642623239548080744038400-33532711560563416561853439568386048000 #1+143843701863402977637624541720094085120 ((#1)^2)+«111»+784949884404592338328674192998635929600000000 ((#1)^77)-66770032635917649300074572955831500800000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,5], ]),(NumericalApproximation[Root, (RowDefault["1.13", FEExpressionsNumericalApproximationElider]), Root[-7483502288449347036028314518617624316932521984+520705090500726388595639300667371906513384767488 #1-17918259245281612121085332629683329410584142675968 ((#1)^2)+«111»+10726271406488796564153071877034428334080000000 ((#1)^77)-246330965949679076990376272921100288000000000 ((#1)^78)+2790015223985432601513046755508224000000000 ((#1)^79)&,1], ])},{0,1},{(NumericalApproximation[Root, (RowDefault["-0.0624", FEExpressionsNumericalApproximationElider]), Root[3348011046054687446588586894387+273675328487397647237991825000783 #1+10528063279784456967398200502468691 ((#1)^2)+255652807673380729611728470237761555 ((#1)^3)+4422420653730204080254904433581059629 ((#1)^4)+«91»+11025799477301561380923592949760 ((#1)^75)-8175821639408563679884718899200 ((#1)^76)-2120259444356145889512456192000 ((#1)^77)+152135800369825007098920960000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,11], ]),(NumericalApproximation[Root, (RowDefault["1.69", FEExpressionsNumericalApproximationElider]), Root[-48784317417568983370796244143031459130638336+1004708778708490899668778250983030047253725184 #1-9472630428891079400640176710181001191623753728 ((#1)^2)+53247068443696535974508410149095909124130996224 ((#1)^3)-192810442891925991053187465168801896784099016704 ((#1)^4)+«104»+36584566337308787146457480728166400 ((#1)^75)-2787180955514696424491990876160000 ((#1)^76)+149893742509030332007786414080000 ((#1)^77)-5094576990025095178380902400000 ((#1)^78)+82521703002365615643033600000 ((#1)^79)&,11], ])},{1,0},{(NumericalApproximation[Root, (RowDefault["0.996", FEExpressionsNumericalApproximationElider]), Root[-202183244354270502812160000+6723099849384182178180115200 #1-88491428008349772009440053440 ((#1)^2)+515341082114902072593104668740 ((#1)^3)-805276325545757050293234984933 ((#1)^4)-6044597943288166479797742666543 ((#1)^5)+«103»+56988002549797880269860510133911552 ((#1)^75)-10289702142507432890642522682949632 ((#1)^76)+1443335983144807204311104288194560 ((#1)^77)-139501386613090879194192150528000 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,11], ]),(NumericalApproximation[Root, (RowDefault["0.0914", FEExpressionsNumericalApproximationElider]), Root[-13399962392802012245702737920000+574188388531566224728362319872000 #1-10852830541366241747978332982476800 ((#1)^2)+119656521932153731581926161199923200 ((#1)^3)-839226726678694541033494735812034560 ((#1)^4)+«106»+2384526529110374491791227098248511488 ((#1)^75)-186930082548111462623483227174600704 ((#1)^76)+10623988549072373660609646742732800 ((#1)^77)-390041502906568093137803988172800 ((#1)^78)+6951636846900945267795689472000 ((#1)^79)&,1], ])},{1,2}}]

Gerbracht 1-11 exact

gs=Table[Symbol["g"<>ToString[i]],{i,11}]

RecognizeGraph/@gs//Union

UnitDistanceGraphEmbeddingQ/@gs//Union

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

Gerbracht additional #1 and #2

Feb 18, 2010 (resent 1/31/2020)

(* Petersen graph *)
(* symmetric outer pentagon *)
(* initial conditions; t as free parameter *) 
xA1=-1/2
yA1=0
xA2=1/2
yA2=0
xA4=0
yA4=t
P01=(xA1-xA5)^2+(yA1-yA5)^2-1
P02=(xA4-xA5)^2+(yA4-yA5)^2-1
P03=(xA2-xA3)^2+(yA2-yA3)^2-1
P04=(xA4-xA3)^2+(yA4-yA3)^2-1

(* maximal t *)
Solve[t^2+1/4==4,t]

S1=Solve[{P01,P02,P03,P04}==0,{xA3,yA3,xA5,yA5}]

S1/.t->1

N[%]

S1[[2]]
N[%/.t->1]

(* Connections with inner Pentagon *)
P05=(xA1-xB1)^2+(yA1-yB1)^2-1
P06=(xA2-xB2)^2+(yA2-yB2)^2-1
P07=(xA3-xB3)^2+(yA3-yB3)^2-1
P08=(xA4-xB4)^2+(yA4-yB4)^2-1
P09=(xA5-xB5)^2+(yA5-yB5)^2-1
(* Inner Star Pentagon *)
P10=(xB1-xB3)^2+(yB1-yB3)^2-1
P11=(xB2-xB4)^2+(yB2-yB4)^2-1
P12=(xB3-xB5)^2+(yB3-yB5)^2-1
P13=(xB4-xB1)^2+(yB4-yB1)^2-1
P14=(xB5-xB2)^2+(yB5-yB2)^2-1

(*  P04 *)
Q01=(xA3-xT1)^2+(yA3-yT1)^2-1
Q02=(xA4-xT1)^2+(yA4-yT1)^2-1
(* P05 *)
Q03=(xA1-xT2)^2+(yA1-yT2)^2-1
Q04=(xB1-xT2)^2+(yB1-yT2)^2-1
(* P14 *)
Q05=(xB2-xT3)^2+(yB2-yT3)^2-1
Q06=(xB5-xT3)^2+(yB5-yT3)^2-1
(* final vertex *)
Q07=(xZ-xT1)^2+(yZ-yT1)^2-1
Q08=(xZ-xT2)^2+(yZ-yT2)^2-1
Q09=(xZ-xT3)^2+(yZ-yT3)^2-1

S2=FindRoot[{P01,P02,P03,P04,P05,P06,P07,P08,P09,P10,P11,P12,P13,P14,Q01,Q02,Q03,Q04,Q05,Q06,Q07,Q08,Q09},
{xA3,0.88272},
{yA3,0.92386},
{t,1.39376},
{xA5,-0.88272},
{yA5,0.92386},
{xB1,0.44624},
{yB1,0.32346},
{xB2,0.36603},
{yB2,0.99099},
{xB3,-0.08659},
{yB3,1.16968},
{xB4,-0.52911},
{yB4,0.54521},
{xB5,0.28354-0.5},
{yB5,0.17815},
{xT1,0.8483},
{yT1,01.92327},
{xT2,0.193-0.5},
{yT2,0.9812},
{xT3,-0.62915},
{yT3,1.08902},
{xZ,.34934-0.5},
{yZ,1.9689},
WorkingPrecision->300]

S3=FindRoot[{P01,P02,P03,P04,P05,P06,P07,P08,P09,P10,P11,P12,P13,P14,Q01,Q02,Q03,Q04,Q05,Q06,Q07,Q08,Q09},
{xA3,1.15-0.5},
{yA3,0.98},
{t,1.7},
{xA5,-0.15-0.5},
{yA5,0.98},
{xB1,.89-0.5},
{yB1,0.45},
{xB2,.99-0.5},
{yB2,0.99},
{xB3,.16-0.5},
{yB3,1.1},
{xB4,.00-0.5},
{yB4,.88},
{xB5,0.42-0.5},
{yB5,.17},
{xT1,1.5-0.5},
{yT1,1.95},
{xT2,.05-0.5},
{yT2,.99},
{xT3,1.4-0.5},
{yT3,.09},
{xZ,1.05-0.5},
{yZ,0.95},
WorkingPrecision->500,MaxIterations->20000]

{P01,P02,P03,P04,P05,P06,P07,P08,P09,P10,P11,P12,P13,P14,Q01,Q02,Q03,Q04,Q05,Q06,Q07,Q08,Q09}/.S3

<<Combinatorica`

Petersen=Graph["Foster010A"];
Heawood=Graph["Foster014A"];

g0=MakeGraph[Table[i,{i,1,10}],{}]
EdA=Table[{i+1,Mod[i+1,5]+1},{i,0,4}]
EdB=Table[{5+i+1,5+Mod[i+2,5]+1},{i,0,4}]
EdAB=Table[{i,5+i},{i,1,5}]
Ed=Union[EdA,EdB,EdAB]
h2=AddEdges[g0,Ed]
H2=MakeUndirected[h2]
GraphPlot[Ed,VertexLabeling->True]

IsomorphicQ[Petersen,H2]

EdH=Union[Complement[Ed,{{1,6},{3,4},{10,7}}],
{{1,11},{6,11},{3,12},{4,12},{7,13},{10,13},{11,14},{12,14},{13,14}}]

g1=MakeGraph[Table[i,{i,1,14}],{}]
h3=AddEdges[g1,EdH]
H3=MakeUndirected[h3]
IsomorphicQ[Heawood,H3]

EH={1->2,1->11,2->3,2->7,3->8,3->12,4->5,4->9,4->12,5->1,5->10,6->8,6->11,7->9,7->13,8->10,9->6,10->13,11->14,12->14,13->14}

VRules=
{1->{xA1,yA1},
2->{xA2,yA2},
3->{xA3,yA3},
4->{xA4,yA4},
5->{xA5,yA5},
6->{xB1,yB1},
7->{xB2,yB2},
8->{xB3,yB3},
9->{xB4,yB4},
10->{xB5,yB5},
11->{xT2,yT2},
12->{xT1,yT1},
13->{xT3,yT3},
14->{xZ,yZ}}

GraphPlot[EH,
VertexCoordinateRules->VRules/.S2]

GraphPlot[EH,
VertexCoordinateRules->VRules/.S3]

Distilled #1

e={{1,2},{1,11},{2,3},{2,7},{3,8},{3,12},{4,5},{4,9},{4,12},{5,1},{5,10},{6,8},{6,11},{7,9},{7,13},{8,10},{9,6},{10,13},{11,14},{12,14},{13,14}};

RecognizeGraph[Graph[e]]

v={{-0.5,0.},{0.5,0.},{-0.310102460407436,0.981803913445394},{0.8795540836588981,0.9251695507191394},{0.3694786055903119,0.9914454930057177},{-0.0896638119563015,1.171374081179576},{0.8518309951404409,1.924785192034839},{0.,1.4009684724486406},{-0.8795540836588981,0.9251695507191394},{-0.528942696340969,0.5523109302164718},{-0.21369132386050338,0.1790953016569798},{0.44521590037497133,0.32644586331937525},{-0.6256222616122524,1.0903103709589337},{-0.14721530268391853,1.9684486184171248}};

h1=IndexGraph[UndirectedEdge@@@e,VertexCoordinates->v]

UnitDistanceGraphEmbeddingQ[h1]

r=RecognizeGraph[h1]

Distilled #2

e={{1,2},{1,11},{2,3},{2,7},{3,8},{3,12},{4,5},{4,9},{4,12},{5,1},{5,10},{6,8},{6,11},{7,9},{7,13},{8,10},{9,6},{10,13},{11,14},{12,14},{13,14}};

RecognizeGraph[Graph[e]]

v={{-0.5,0.},{0.5,0.},{-0.4489025646564063,0.9986936728052838},{0.6433279669087985,0.9896752466853914},{0.48626807076295514,0.9999057126146589},{-0.3466517666585787,1.1308847616830182},{0.9846849825522739,1.9296089613009382},{0.,1.7552659513004567},{-0.6433279669087985,0.9896752466853914},{-0.5081049967582191,0.8939708147361485},{-0.07105650596632446,0.16961099615939362},{0.3904428089199568,0.45509515932685696},{0.9266620994765724,0.10210111278606053},{0.5506467078280182,1.0287145301524314}};

h2=IndexGraph[UndirectedEdge@@@e,VertexCoordinates->v]

UnitDistanceGraphEmbeddingQ[h2]

r=RecognizeGraph[h2]

Join[GraphData[r,"Embeddings"],ToCommonEdges[#,GraphData@r]&/@{h1,h2}]

GetGraphData[r,"File"]

Gerbracht Hexagon

Import["09-12-31-Heawood-hex.pdf"]

g=;

g2=DeleteCases[g,{___,_Dashing,___}|_CapForm,{1,∞}]

pts=Cases[g2,HoldPattern[BezierCurve[x_,SplineClosed->True,SplineDegree->3]]:>Mean[x],{1,∞}]

Length[pts]

Graphics[{{Red,PointSize[.02],Point[pts]},MapIndexed[Text[#2[[1]],#1+10]&,pts]},Axes->True]

s=Norm[Subtract@@pts[[{3,7}]]]

pts2=Delete[pts,{{15},{13}}]/s

Length[pts2]

Graphics[{{Red,PointSize[.02],Point[pts2]},MapIndexed[Text[#2[[1]],#1+.1]&,pts2]},Axes->True]

Norm[Subtract@@pts2[[{3,7}]]]

e=Select[Subsets[Range[14],{2}],.98<Norm[Subtract@@pts2[[#]]]<1.02&]

Length[%]

ShowLabeledGraph[g=Graph[List/@e,List/@pts2]]

Norm[Subtract@@pts2[[#]]]&/@e

ToCommonEdges[g,"HeawoodGraph"]

GetGraphData["HeawoodGraph","File"]

GraphPlot[g,Method->None,ImageSize->200]

Horvat horizontally symmetric

Horvat

http://eprints.fri.uni-lj.si/858/1/horvat_boris.disertacija.pdf

Import["/Users/eww/Downloads/horvat_boris.disertacija.pdf",{"Pages",68}]

v={{73.5`,26.5`},{88.5`,31.5`},{241.5`,52.5`},{260.5`,32.5`},{381.5`,152.5`},{328.5`,201.5`},{282.5`,202.5`},{158.5`,187.5`},{108.5`,195.5`},{30.5`,193.5`},{150.5`,315.5`},{155.5`,360.5`},{312.5`,371.5`},{322.5`,313.5`}};

ShowLabeledGraph[Graph[{},List/@v]]

Norm[Subtract@@v[[#]]]&/@{{1,10},{2,10}}

v2=v/172.2

Norm[Subtract@@v2[[#]]]&/@{{1,10},{2,10}}

e=Select[Subsets[Range[Length[v]],{2}],.98<Norm[Subtract@@v2[[#]]]<1.02&]//Quiet

Length[%]

GraphData["HeawoodGraph","E"]

ShowLabeledGraph[g=Graph[List/@e,List/@v2]]

e2=DeleteCases[e,{7,11}];

ShowLabeledGraph[g=Graph[List/@e2,List/@v2]]

RecognizeGraph[g]

Norm/@Subtract@@@(Vertices[g][[#]]&/@Edges[g])

pairs=Table[{x[i],y[i]},{i,14}]

eqns=((x[#[[1]]]-x[#[[2]]])^2+(y[#[[1]]]-y[#[[2]]])^2-1)&/@e2

Length[eqns]

vars=List@@@Flatten[Thread/@Thread[pairs->Vertices[g]]]

NSolve[Evaluate[eqns],Evaluate[Flatten[pairs]]]

FindRoot[Evaluate[eqns],Evaluate[vars]]

FindMinimum

(soln=FindMinimum[{Total[Abs[eqns]],Thread[eqns==0]},vars])//Timing

ShowGraph[g2=ChangeVertices[g,pairs/.soln[[2]]]]

GraphPlot[g2,Method->None,ImageSize->200]

Norm/@Subtract@@@(Vertices[g2][[#]]&/@Edges[g2])

ToCommonEdges[g2,"HeawoodGraph"]

NMinimize

(soln=NMinimize[{Total[Abs[eqns]],Thread[eqns==0]},vars[[All,1]],Method->{Automatic,"InitialPoints"->vars[[All,2]]}])//Timing

ShowGraph[g=ChangeVertices[g,pairs/.soln[[2]]]]

Norm/@Subtract@@@(Vertices[g][[#]]&/@Edges[g])

(soln=NMinimize[{Total[Abs[eqns]],Thread[eqns==0]},vars[[All,1]],Method->{"RandomSearch","InitialPoints"->Evaluate[vars[[All,2]]]}])//Timing