The independent domination number of maximal triangle-free graphs

A triangle-free graph is maximal if the addition of any edge produces a triangle. A set S of vertices in a graph G is called an independent dominating set if S is both an independent and a dominating set of G. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. In this paper, we show that i(G) ≤ δ(G) ≤ n 2 for maximal triangle-free graphs G of order n and minimum degree δ(G). We characterize the graphs attaining the latter bound. We also show that, given a positive integer k ≥ 2 and any positive integer n ≥ 5k 2 , there exists a non-bipartite maximal triangle-free graph G of order n with i(G) = k.