On the Number of Minimal Dominating Sets on Cobipartite and Interval Graphs
Résumé
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(1.7159^n) minimal dominating sets. Here we establish upper bounds on this maximum number of minimal dominating sets for cobipartite and interval graphs. For each of these graph classes, we provide an algorithm to enumerate them. For interval graphs, we show that the number of minimal dominating sets is at most 3^{n/3} \approx 1.4423^n, which is the best possible bound. For cobipartite graphs, we lower the O(1.5875^n) upper bound from Couturier et al. to O(1.4511^n).