On the number of minimal dominating sets on some graph classes

A dominating set in a graph is a subset of vertices such that each vertex is either in the dominating set or adjacent to some vertex in the dominating set. It is known that graphs have at most O(1.7159n)O(1.7159n) minimal dominating sets. In this paper, we establish upper bounds on this maximum number of minimal dominating sets for split graphs, cobipartite graphs and interval graphs. For each of these graph classes, we provide an algorithm to enumerate them. For split and interval graphs, we show that the number of minimal dominating sets is at most 3n/3≈1.4423n3n/3≈1.4423n, which is the best possible bound. This settles a conjecture by Couturier et al. (SOFSEM 2012, [1]). For cobipartite graphs, we lower the O(1.5875n)O(1.5875n) upper bound from Couturier et al. to O(1.4511n)O(1.4511n).