Counting dominating sets in some subclasses of bipartite graphs

Counting dominating sets in a graph is a #P-complete problem even for bipartite graphs. This paper concerns the class of tree convex bipartite graphs, which is a subclass of bipartite graphs. A bipartite graph G with bipartition (X,Y) is called tree convex, if a tree T defined on X exists such that for every vertex y in Y, the neighbors of y form a subtree in T. For each y in Y, when the neighbors of y form a path or a rooted directed path in T, the tree convex bipartite graph is called path-tree bipartite or rooted path-tree bipartite, respectively. This work proves that the problem of counting dominating sets remains #P-complete for path-tree bipartite graphs but admits a polynomial-time solution for the more tightly constrained subclass of rooted path-tree bipartite graphs.