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.

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 .

See also

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

Explore with Wolfram|Alpha

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

Independent Set

Referenced on Wolfram|Alpha

Independent Vertex Set

Cite this as:

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

Subject classifications