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# Independent Vertex Set

An independent vertex set of a graph , also known as a stable set, is a subset of the vertices such that no two vertices in the subset represent an edge of . The figure above shows independent sets consisting of two subsets for a number of graphs (the wheel graph , utility graph , Petersen graph, and Frucht graph).

Any independent vertex set is an irredundant set (Burger et al. 1997, Mynhardt and Roux 2020).

The polynomial whose coefficients give the number of independent vertex sets of each cardinality in a graph is known as its independence polynomial.

A set of vertices is an independent vertex set iff its complement forms a vertex cover (Skiena 1990, p. 218). The counts of vertex covers and independent vertex sets in a graph are therefore the same.

The empty set is trivially an independent vertex set since it contains no vertices, and therefore no edge endpoints.

A maximum independent vertex set is an independent vertex set of a graph containing the largest possible number of vertices for the given graph, and the cardinality of this set is called the independence number of the graph.

An independent vertex set that cannot be enlarged to another independent vertex set by adding a vertex is called a maximal independent vertex set.

In the Wolfram Language, the command FindIndependentVertexSet[g][[1]] can be used to find a maximum independent vertex set, and FindIndependentVertexSet[g, Length /@ FindIndependentVertexSet[g], All] to find all maximum independent vertex sets. Similarly, FindIndependentVertexSet[g, Infinity] can be used to find a maximal independent vertex set, and FindIndependentVertexSet[g, Infinity, All] to find all independent vertex sets. To find all independent vertex sets in the Wolfram Language, enumerate all vertex subsets and select those for which IndependentVertexSetQ[g, s] is true.

Independent vertex set counts for some families of graphs are summarized in the following table.

 graph family OEIS number independent vertex sets antiprism graph for A000000 X, X, 10, 21, 46, 98, 211, 453, 973, 2090, ... bishop graph A201862 X, 9, 70, 729, 9918, 167281, ... black bishop graph A000000 X, X, X, 27, 114, 409, 2066, ... -folded cube graph A000000 X, 3, 5, 31, 393, ... grid graph for A006506 X, 7, 63, 1234, 55447, 5598861, ... grid graph for A000000 X, 35, 70633, ... -halved cube graph A000000 2, 3, 5, 13, 57, ... -Hanoi graph A000000 4, 52, 108144, ... hypercube graph A027624 3, 7, 35, 743, 254475, 19768832143, ... king graph A063443 X, 5, 35, 314, 6427, ... knight graph A141243 X, 16, 94, 1365, 55213, ... -Möbius ladder A182143 X, X, 15, 33, 83, 197, 479, 1153, 2787, ... -Mycielski graph A000000 2, 3, 11, 103, 7407, ... odd graph A000000 2, 4, 76, ... prism graph for A051927 X, X, 13, 35, 81, 199, 477, 1155, 2785, ... queen graph A000000 2, 5, 18, 87, 462, ... rook graph A002720 2, 7, 34, 209, 1546, 13327, 130922, ... -Sierpiński gasket graph A000000 4, 14, 440, ... -triangular graph A000000 X, 2, 4, 10, 26, 76, 232, 764, ... -web graph for A000000 X, X, 68, 304, 1232, 5168, 21408, ... white bishop graph A000000 X, X, X, 27, 87, 409, 1657, ...

Many families of graphs have simple closed forms for counts of independent vertex sets, as summarized in the following table. Here, is a Fibonacci number, is a Lucas number, is a Laguerre polynomial, is the golden ratio, and , , are the roots of .

Clique, Disjoint Sets, Edge Cover, Empty Set, Independence Number, Independence Polynomial, Independent Set, Maximal Independent Vertex Set, Maximum Independent Edge Set, Maximum Independent Set Problem, Maximum Independent Vertex Set, Venn Diagram, Vertex Cover

## References

Burger, A. P.; Cockayne, E. J.; and Mynhardt, C. M. "Domination and Irredundance in the Queens' Graph." Disc. Math. 163, 47-66, 1997.Gallai, T. "Über extreme Punkt- und Kantenmengen." Ann. Univ. Sci. Budapest, Eőtvős Sect. Math. 2, 133-138, 1959.Hochbaum, D. S. (Ed.). Approximation Algorithms for NP-Hard Problems. PWS Publishing, p. 125, 1997.Mynhardt, C. M. and Roux, A. "Irredundance Graphs." 14 Apr. 2020. https://arxiv.org/abs/1812.03382.Myrvold, W. and Fowler, P. W. "Fast Enumeration of All Independent Sets up to Isomorphism." J. Comb. Math. Comb. Comput. 85, 173-194, 2013.Skiena, S. "Maximum Independent Set" §5.6.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 218-219, 1990.

SeeAlso

## Cite this as:

Weisstein, Eric W. "Independent Vertex Set." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IndependentVertexSet.html