Maximal Independent Vertex Set

A maximal independent vertex set of a graph is an independent vertex set that cannot be expanded to another independent vertex set by addition of any vertex in the graph.

A maximal independent vertex set of a graph  is equivalent to a maximal clique on the graph complement .

Note that a maximal independent vertex set is not equivalent to a maximum independent vertex set, which is an independent vertex set containing the largest possible number of vertices among all independent vertex sets. A maximum independent vertex set is always maximal, but the converse does not hold.

A subset  of the vertex set  of a graph is a maximally independent vertex set iff  is both a dominating set and an independent vertex set (Burger et al. 1997).

Any maximal independent vertex set is also both minimal dominating and maximal irredundant (Mynhardt and Roux 2020).

A maximal independent vertex set of a graph can be computed in the Wolfram Language using FindIndependentVertexSet[g, Infinity], and all maximal independent vertex sets can be computed using FindIndependentVertexSet[g, Infinity, All].

REFERENCES

Burger, A. P.; Cockayne, E. J.; and Mynhardt, C. M. "Domination and Irredundance in the Queens' Graph." Disc. Math. 163, 47-66, 1997.

Hedetniemi, S. T. and Laskar, R. C. "A. Bibliography on Dominating Sets in Graphs and Some Basic Definitions of Domination Parameters." Disc. Math. 86, 257-277, 1990.

Myrvold, W. and Fowler, P. W. "Fast Enumeration of All Independent Sets up to Isomorphism." J. Comb. Math. Comb. Comput. 85, 173-194, 2013.