Minimal dominating sets in graph classes: Combinatorial bounds and enumeration

The number of minimal dominating sets that a graph on n vertices can have is known to be at most 1.7159n. This upper bound might not be tight, since no examples of graphs with 1.5705n or more minimal dominating sets are known. For several classes of graphs, we substantially improve the upper bound on the number of minimal dominating sets. At the same time, we give algorithms for enumerating all minimal dominating sets, where the running time of each algorithm is within a polynomial factor of the proved upper bound for the graph class in question. In several cases, we provide examples of graphs containing the maximum possible number of minimal dominating sets for graphs in that class, thereby showing the corresponding upper bounds to be tight.