Fractional Independence Number

Author

Eric W. Weisstein
December 7, 2023

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

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

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

GraphDataCountTable

GraphDataCountTable["FractionalIndependenceNumber"]

Parametrized families

<<MathWorld`Graphs`

Alkane

seq=Table[GraphData[{"Alkane",n},"FractionalIndependenceNumber"],{n,10}]

FindSequenceFunction[seq,n]

AlternatingGroupGraph

seq=Table[GraphData[{"AlternatingGroupGraph",n},"FractionalIndependenceNumber"],{n,2,6}]

Andrasfai

seq=Table[GraphData[{"Andrasfai",n},"FractionalIndependenceNumber"],{n,10}]

FindSequenceFunction[seq,n]

Antelope n×n

seq=Table[GraphData[{"Antelope",{n,n}},"FractionalIndependenceNumber"],{n,15}]

Antiprism

Entity["Graph",{"Antiprism",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Antiprism",n},"FractionalIndependenceNumber"],{n,3,25}]

FindSequenceFunction[seq,n-2]

Apollonian

seq=Table[GraphData[{"Apollonian",n},"FractionalIndependenceNumber"],{n,5}]

FindSequenceFunction[seq,n]

Barbell

seq=Table[GraphData[{"Barbell",n},"FractionalIndependenceNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]

BlackBishop n×n

seq=Table[GraphData[{"BlackBishop",{n,n}},"FractionalIndependenceNumber"],{n,10}]

Bishop n×n

seq=Table[GraphData[{"Bishop",{n,n}},"FractionalIndependenceNumber"],{n,10}]

Book

seq=Table[GraphData[{"Book",n},"FractionalIndependenceNumber"],{n,20}]

FindSequenceFunction[seq,n]

Bruhat

seq=Table[GraphData[{"Bruhat",n},"FractionalIndependenceNumber"],{n,6}]

FindSequenceFunction[seq,n]

Camel n×n

seq=Table[GraphData[{"Camel",{n,n}},"FractionalIndependenceNumber"],{n,10}]

Centipede

seq=Table[GraphData[{"Centipede",n},"FractionalIndependenceNumber"],{n,10}]

FindSequenceFunction[seq,n]

CocktailParty

seq=Table[GraphData[{"CocktailParty",n},"FractionalIndependenceNumber"],{n,20}]

Complete

Entity["Graph",{"Complete",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Complete",n},"FractionalIndependenceNumber"],{n,30}]

FindSequenceFunction[seq,n]

CompleteBipartite m×n

Entity["Graph",{"CompleteBipartite",{m,n}}]["FractionalIndependenceNumber"]

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"CompleteBipartite",{m,n}},"FractionalIndependenceNumber"],
Max[m,n]
},{n,20},{m,n}],1],{_,x_,x_}]

CompleteBipartite n×n

Entity["Graph",{"CompleteBipartite",{n,n}}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"CompleteBipartite",{n,n}},"FractionalIndependenceNumber"],{n,20}]

FindSequenceFunction[seq,n]//FullSimplify

CompleteTripartite k×m×n

Entity["Graph",{"CompleteTripartite",{k,m,n}}]["FractionalIndependenceNumber"]

DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"FractionalIndependenceNumber"],
Piecewise[{{Max[k,m,n],k+m+n<2Max[k,m,n]}},(k+m+n)/2]
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CompleteTripartite n×n×n

Entity["Graph",{"CompleteTripartite",{n,n,n}}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"CompleteTripartite",{n,n,n}},"FractionalIndependenceNumber"],{n,10}]

FindSequenceFunction[seq,n]//FullSimplify

CrossedPrism

seq=Table[GraphData[{"CrossedPrism",2n},"FractionalIndependenceNumber"],{n,2,10}]

FindSequenceFunction[seq,n-1]//Factor

Crown

seq=Table[GraphData[{"Crown",n},"FractionalIndependenceNumber"],{n,20}]

CubeConnectedCycle

Entity["Graph",{"CubeConnectedCycle",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"CubeConnectedCycle",n},"FractionalIndependenceNumber"],{n,3,8}]

FindSequenceFunction[seq,n-2]

Cycle

Entity["Graph",{"Cycle",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Cycle",n},"FractionalIndependenceNumber"],{n,3,30}]

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

CycleComplement

seq=Table[GraphData[{"CycleComplement",n},"FractionalIndependenceNumber"],{n,3,20}]

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

DiagonalIntersection

seq=Table[GraphData[{"DiagonalIntersection",n},"FractionalIndependenceNumber"],{n,3,14}]

Dipyramid

seq=Table[GraphData[{"Dipyramid",n},"FractionalIndependenceNumber"],{n,3,20}]

DoubleCone

seq=Table[GraphData[{"DoubleCone",n},"FractionalIndependenceNumber"],{n,3,20}]

Empty

Entity["Graph",{"Empty",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Empty",n},"FractionalIndependenceNumber"],{n,30}]

FibonacciCube

A173173

seq=Table[GraphData[{"FibonacciCube",n},"FractionalIndependenceNumber"],{n,15}]

FullSimplify[FindSequenceFunction[seq,n],n∈Integers]

Table[Ceiling[Fibonacci[n+2]/2],{n,15}]

Flower

seq=Table[GraphData[{"Flower",n},"FractionalIndependenceNumber"],{n,5,20}]

FoldedCube

seq=Table[GraphData[{"FoldedCube",n},"FractionalIndependenceNumber"],{n,2,10}]

FindSequenceFunction[seq,n-1]

Gear

seq=Table[GraphData[{"Gear",n},"FractionalIndependenceNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]

Giraffe n×n

seq=Table[GraphData[{"Giraffe",{n,n}},"FractionalIndependenceNumber"],{n,20}]

Goldberg

seq=Table[GraphData[{"Goldberg",n},"FractionalIndependenceNumber"],{n,3,20}]

Grid m×n

Entity["Graph",{"Grid",{m,n}}]["FractionalIndependenceNumber"]

DeleteCases[Sort@Flatten[Table[{
{m,n},
GraphData[{"Grid",{m,n}},"FractionalIndependenceNumber"],
Ceiling[m n/2]
},{n,10},{m,1}],1],{_,x_,x_}]

Grid n×n

A000982

Entity["Graph",{"Grid",{n,n}}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Grid",{n,n}},"FractionalIndependenceNumber"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

Table[Ceiling[n^2/2],{n,20}]

Grid k×m×n

Entity["Graph",{"Grid",{k,m,n}}]["FractionalIndependenceNumber"]

DeleteCases[Sort@Flatten[Table[{
{k,m,n},
GraphData[{"Grid",{k,m,n}},"FractionalIndependenceNumber"],
Ceiling[k m n/2]
},{n,5},{m,n},{k,m}],2],{_,x_,x_}]

Grid n×n×n

A036486

Entity["Graph",{"Grid",{n,n,n}}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Grid",{n,n,n}},"FractionalIndependenceNumber"],{n,10}]

Table[Ceiling[n^3/2],{n,10}]

HalvedCube

seq=Table[GraphData[{"HalvedCube",n},"FractionalIndependenceNumber"],{n,10}]

FindSequenceFunction[Rest@seq,n-1]

Hanoi

seq=Table[GraphData[{"Hanoi",n},"FractionalIndependenceNumber"],{n,5}]

FindSequenceFunction[seq,n]

Helm

seq=Table[GraphData[{"Helm",n},"FractionalIndependenceNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]

Hypercube

Entity["Graph",{"Hypercube",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Hypercube",n},"FractionalIndependenceNumber"],{n,0,10}]

FindSequenceFunction[Rest@seq,n]

Keller

Entity["Graph",{"Keller",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Keller",n},"FractionalIndependenceNumber"],{n,5}]

FindSequenceFunction[Rest@seq,n-1]//Simplify

Table[(2^(2n-1)),{n,5}]

King n×n

seq=Table[GraphData[{"King",{n,n}},"FractionalIndependenceNumber"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//Factor

Knight m×n

Entity["Graph",{"Knight",{m,n}}]["FractionalIndependenceNumber"]

DeleteCases[Sort@Flatten[Table[{
{m,n},
GraphData[{"Knight",{m,n}},"FractionalIndependenceNumber"],
Piecewise[{{Max[m,n],Min[m,n]==1},{1+Max[m,n]-Cos[((Max[m,n] π)/2)],Min[m,n]==2}},Ceiling[m n/2]]
},{n,10},{m,n}],1],{_,x_,x_}]

seq=Table[FractionalIndependenceNumber[KnightGraph[2,n]],{n,40}]

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n∈Integers&&n≥0]

Knight n×n

A000982

Entity["Graph",{"Knight",{n,n}}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Knight",{n,n}},"FractionalIndependenceNumber"],{n,20}]

FindSequenceFunction[Drop[seq,3],n-3]//FullSimplify

Table[Ceiling[((n^2)/2)],{n,20}]

Ladder

Entity["Graph",{"Ladder",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Ladder",n},"FractionalIndependenceNumber"],{n,20}]

FindSequenceFunction[seq,n]//Factor

LadderRung

seq=Table[GraphData[{"LadderRung",n},"FractionalIndependenceNumber"],{n,20}]

LucasCube

A364587

seq=Table[GraphData[{"LucasCube",n},"FractionalIndependenceNumber"],{n,15}]

FindSequenceFunction[seq,n]//FullSimplify

Table[(4+3 LucasL[n]+2 Cos[2 n Pi/3])/6,{n,20}]

MengerSponge

seq=Table[GraphData[{"MengerSponge",n},"FractionalIndependenceNumber"],{n,3}]

FindSequenceFunction[seq,n]

MoebiusLadder

Entity["Graph",{"MoebiusLadder",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"MoebiusLadder",n},"FractionalIndependenceNumber"],{n,3,25}]

FindSequenceFunction[seq,n-2]

Mycielski

Entity["Graph",{"Mycielski",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Mycielski",n},"FractionalIndependenceNumber"],{n,11}]

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

Odd

Entity["Graph",{"Odd",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Odd",n},"FractionalIndependenceNumber"],{n,7}]

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

Table[2^n Pochhammer[(5/2),-2+n]/4,{n,10}]

Table[(2(n-2)+3)!!/3,{n,10}]

Table[4((3 (4^(-2+n)) (2(n-2)+3)!!/3)/((2^n)Gamma[1+n])),{n,10}]

FullSimplify[4((3 (4^(-2+n)) (2(n-2)+3)!!/3)/((2^n)Gamma[1+n]))]

Table[(((2^(n-2)) (2 n-1)!!)/(n!)),{n,10}]

FullSimplify[(((2^(n-2)) (2 n-1)!!)/(n!))]

Table[(((2^(n-2)) (2 n-1)!!)/(n!)),{n,10}]

Pan

seq=Table[GraphData[{"Pan",n},"FractionalIndependenceNumber"],{n,3,20}]

Path

Entity["Graph",{"Path",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Path",n},"FractionalIndependenceNumber"],{n,20}]

FullSimplify[FindSequenceFunction[Rest@seq,n-1],n∈Integers]

Table[Ceiling[n/2],{n,20}]

PathComplement

seq=Table[GraphData[{"PathComplement",n},"FractionalIndependenceNumber"],{n,20}]

FullSimplify[FindSequenceFunction[Drop[seq,3],n-3],n∈Integers]

Pell

A113225

seq=Table[GraphData[{"Pell",n},"FractionalIndependenceNumber"],{n,9}]

FindSequenceFunction[seq,n]//FullSimplify

Table[Ceiling[Fibonacci[n+1,2]/2],{n,9}]

Prism

Entity["Graph",{"Prism",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Prism",n},"FractionalIndependenceNumber"],{n,3,25}]

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

Queen n×n

seq=Table[GraphData[{"Queen",{n,n}},"FractionalIndependenceNumber"],{n,20}]

FullSimplify[FindSequenceFunction[Rest@seq,n-1],n∈Integers]//Factor

Rook n×n

seq=Table[GraphData[{"Rook",{n,n}},"FractionalIndependenceNumber"],{n,20}]

FullSimplify[FindSequenceFunction[Rest@seq,n-1],n∈Integers]//Factor

RookComplement n×n

seq=Table[GraphData[{"RookComplement",{n,n}},"FractionalIndependenceNumber"],{n,10}]

FindSequenceFunction[Drop[seq,2],n-2]//Factor

SierpinskiCarpet

seq=Table[GraphData[{"SierpinskiCarpet",n},"FractionalIndependenceNumber"],{n,4}]

FindSequenceFunction[seq,n]

SierpinskiGasket

seq=Table[GraphData[{"SierpinskiGasket",n},"FractionalIndependenceNumber"],{n,5}]

FindSequenceFunction[seq,n]

SierpinskiTetrahedron

seq=Table[GraphData[{"SierpinskiTetrahedron",n},"FractionalIndependenceNumber"],{n,6}]

FindSequenceFunction[seq,n]

Spoke n×n

seq=Table[GraphData[{"Spoke",{n,n}},"FractionalIndependenceNumber"],{n,5}]

FindSequenceFunction[seq,n]

StackedBook

seq=Table[GraphData[{"StackedBook",{n,n}},"FractionalIndependenceNumber"],{n,10}]

StackedPrism

seq=Table[GraphData[{"StackedPrism",{n,n}},"FractionalIndependenceNumber"],{n,3,5}]

Star

Entity["Graph",{"Star",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Star",n},"FractionalIndependenceNumber"],{n,25}]

FindSequenceFunction[Rest@seq,n-1]//Factor

Sun

seq=Table[GraphData[{"Sun",n},"FractionalIndependenceNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Factor

Sunlet

seq=Table[GraphData[{"Sunlet",n},"FractionalIndependenceNumber"],{n,3,20}]

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

TorusGrid n×n

seq=Table[GraphData[{"TorusGrid",{n,n}},"FractionalIndependenceNumber"],{n,3,10}]

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

Transposition

Entity["Graph",{"Transposition",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Transposition",n},"FractionalIndependenceNumber"],{n,6}]

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

FunctionExpand[%]

Table[n!/2,{n,6}]

Trapezohedron

Entity["Graph",{"Trapezohedron",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Trapezohedron",n},"FractionalIndependenceNumber"],{n,3,20}]

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

Triangular

seq=Table[GraphData[{"Triangular",n},"FractionalIndependenceNumber"],{n,2,20}]

FindSequenceFunction[Rest@seq,n-2]//FullSimplify

TriangularGrid

seq=Table[GraphData[{"TriangularGrid",n},"FractionalIndependenceNumber"],{n,0,10}]

FindSequenceFunction[Rest@seq,n]//FullSimplify

TriangularHoneycombAcuteKnight

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"FractionalIndependenceNumber"],{n,20}]

TriangularHoneycombBishop

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"FractionalIndependenceNumber"],{n,9}]

FindSequenceFunction[seq,n]//Factor

TriangularHoneycombObtuseKnight

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"FractionalIndependenceNumber"],{n,20}]

TriangularHoneycombQueen

seq=Table[GraphData[{"TriangularHoneycombQueen",n},"FractionalIndependenceNumber"],{n,10}]

FindSequenceFunction[seq,n]

TriangularHoneycombRook

seq=Table[GraphData[{"TriangularHoneycombRook",n},"FractionalIndependenceNumber"],{n,10}]

TriangularSnake

seq=Table[GraphData[{"TriangularSnake",n},"FractionalIndependenceNumber"],{n,1,20,2}]

Web

seq=Table[GraphData[{"Web",n},"FractionalIndependenceNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

Wheel

Entity["Graph",{"Wheel",n}]["FractionalIndependenceNumber"]

seq=Table[GraphData[{"Wheel",n},"FractionalIndependenceNumber"],{n,4,20}]

FindSequenceFunction[seq,n-3]//FullSimplify

WheelComplement

seq=Table[GraphData[{"WheelComplement",n},"FractionalIndependenceNumber"],{n,4,20}]

FindSequenceFunction[seq,n-3]//FullSimplify

WhiteBishop n×n

A002620

seq=Table[GraphData[{"WhiteBishop",{n,n}},"FractionalIndependenceNumber"],{n,2,10}]

FullSimplify[FindSequenceFunction[Rest@seq,n-2],n∈Integers]

Table[Floor[n^2/4],{n,2,10}]

Zebra n×n

seq=Table[GraphData[{"Zebra",{n,n}},"FractionalIndependenceNumber"],{n,20}]

FindSequenceFunction[Drop[seq,5],n-5]//Timing

gf=(x^5)FindGeneratingFunction[Drop[seq,5],x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>8]

Algorithms

<<MathWorld`Graphs`

Disconnected

GraphData["K1","IndependenceNumber"]

FindFractionalIndependentVertexSet[GraphData["2K1"]]

FindFractionalIndependentVertexSet[GraphData["K1+K2"]]

FindFractionalIndependentVertexSet[GraphData["2C3"]]

FindFractionalIndependentVertexSet[GraphData["P2+C3"]]

Total[%]

FindFractionalIndependentVertexSet[GraphData[#]]&/@{"P2","C3"}

Total/@%

Total[%]

<<MathWorld`Graphs`

DeleteCases[Function[g,
{
g,
Total[GraphData[#,"FractionalIndependenceNumber"]&/@GraphData[g,"ConnectedComponents","Name"]],
FractionalIndependenceNumber[GraphData[g]]
}]/@GraphData["Disconnected",;;7],{_,x_,x_}]

Automatic

FindFractionalIndependentVertexSet[CycleGraph[3]]

(l1=Monitor[{gn=#,FractionalIndependenceNumber[GraphData[#]]//Timing}&/@GraphData["Connected",21;;31],gn])//Timing

"MOSEK"

FindFractionalIndependentVertexSet[CycleGraph[3],Method->"MOSEK"]//InputForm

(l2=Monitor[{gn=#,FractionalIndependenceNumber[GraphData[#],Method->"MOSEK"]//Timing}&/@GraphData["Connected",21;;31],gn])//Timing

l1[[All,2,2]]===l2[[All,2,2]]

Examples

Select[GraphData["Properties"],GraphData[{"Wheel",5},#]==5/2&]

Cases[GraphData[{"Wheel",5},#]&/@GraphData["Properties"],5/2]

Select[GraphData["Properties"],GraphData[{"Wheel",5},#]==5/2&&GraphData[{"Complete",5},#]==5/2&]

"SingletonGraph"

FindIndependentVertexSet[CompleteGraph[1]]

FindIndependentVertexSet[Graph[{1},{}]]

FindIndependentVertexSet[Graph[{1,2},{}]]

Special case; make it equal to α:

GraphData["SingletonGraph","IndependenceNumber"]

FindFractionalIndependentVertexSet[CompleteGraph[1]]

GraphData["SingletonGraph","FractionalIndependenceNumber"]

(SuperStar[α])((Subscript[W, n]))==(n/2)

FindFractionalIndependentVertexSet[WheelGraph[4]]

FindFractionalIndependentVertexSet[WheelGraph[5]]

(SuperStar[α])((Subscript[K, n]))==(n/2)

FindFractionalIndependentVertexSet[CompleteGraph[3]]

FindFractionalIndependentVertexSet[CompleteGraph[4]]

FindFractionalIndependentVertexSet[CompleteGraph[5]]

Willis 2011, p. 4

Fig. 1.1

v={{273.81378487518356`,75.16060939794409`},{340.38064427312776`,348.368667400881`},{476.63553903573177`,210.80557085168857`},{409.42088839941266`,209.19029919236414`},{339.825394640235`,210.7508871757219`},{272.3499449339207`,211.1715308370043`},{206.39301884483604`,210.34286282427797`},{139.57377325012237`,208.2017865883504`}}/476;

g=Graph[Range@Length@v,UndirectedEdge@@@{e[1,3,2,8,1,6],e[8,7,6,5,4,3],{2,5}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

FindFractionalIndependentVertexSet[g]

Total[%]

Fig 2.6

v={{556.4332667951542`,507.4596591803015`},{504.49218750000006`,561.3220283983633`},{357.3960627753304`,509.67004601950856`},{406.71720126651985`,559.5874469005659`},{459.242497246696`,712.116184510698`},{557.8847742290749`,756.5889268014469`},{356.29388078193836`,756.3299441472618`}}/757;

v={{1,0},{3/4,1/3},{0,0},{1/4,1/3},{1/2,2/3},{1,1},{0,1}};

g=Graph[Range@Length@v,UndirectedEdge@@@{e[4,3,1,6,7,3,2,5,4,1,2],e[7,5,6]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

RectilinearCrossingCount@g

GraphData[{7,763},"Graph","All"]

FindFractionalIndependentVertexSet[g]

Total[%]

Fig 2.7

v={{501.4832277900148`,248.59080012140646`},{414.41503762848754`,301.28571109791164`},{329.8702505506608`,250.7367711199381`},{417.45066997063145`,248.39130030496005`},{416.9923595814978`,147.74633885121557`},{501.0303092878121`,129.90997688352104`},{330.3986554698972`,130.51386821979122`}}/501.;

v={#,1-#2}&@@@{{1,1},{1/2,1},{0,1},{1/2,3/4},{1/2,1/4},{1,0},{0,0}};

g=Graph[Range@Length@v,UndirectedEdge@@@{e[7,6,1,2,3,7,5,6,4,7],e[6,3,5,1,7],e[5,4,2]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

DegenerateGraphEmbeddingTypes[g]

GraphData[%,"Graph","All"]

FindFractionalIndependentVertexSet[g]

Total[%]