Independent dominating sets in graphs of girth five

Let gamma(G) and gamma(omicron)(G) denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then gamma(omicron)(G) <= n/d (log d + 1). In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then gamma(omicron)(G) <= n/d (log d + c) for some constant c independent of d. This result is sharp in the sense that as d ->infinity, almost all d-regular n-vertex graphs G of girth at least five have gamma(omicron)(G) similar to n/d log d. Furthermore, if G is a disjoint union of n/(2d) complete bipartite graphs K-d,K-d, then gamma(omicron)(G)=n/2. We also prove that there are n-vertex graphs G of minimum degree d and whose maximum degree grows notmuch faster than d log d such that gamma(omicron)(G) similar to n/2 as d ->infinity. Therefore both the girth and regularity conditions are required for the main result.