Efficient enumeration of dominating sets for sparse graphs

Abstract

A dominating set D of a graph G is a set of vertices such that any vertex in G is in D or its neighbor is in D. Enumeration of minimal dominating sets in a graph is one of the central problems in enumeration study since enumeration of minimal dominating sets corresponds to the enumeration of minimal hypergraph transversals. The output-polynomial time enumeration of minimal hypergraph transversals is an interesting open problem. On the other hand, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a k-degenerate graph in Ok time per solution using On+m space, where n and m are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, H-minor free graphs with some fixed H. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine.