The Complexity of Some Problems on Maximal Independent Sets in Graphs

Let mi(G) be the number of maximal independent sets in a graph G. A graph G is mi-minimal if mi(H) < mi(G) for each proper induced subgraph H of G. As it is shown in [6], every graph G without duplicated or isolated vertices has at most 2k-1 +k - 2 vertices, where k = mi(G) > 2. Hence the extremal problem of calculating m(k) = max{IV(G)1: G is a mi-minimal graph with mi(G) = k} has a solution for any k ~ 1 We show that 2(k -1) ~ m(k) ~ k(k -1) for any k ~ andconjecture that m(k) = 2(k - 1). We also prove NP-completeness of some related problems.