Counting independent sets and maximal independent sets in some subclasses of bipartite graphs

The problems of counting independent sets and maximal independent sets are #P-complete for tree convex bipartite graphs but solvable in polynomial time for its subclass of convex bipartite graphs. This study investigates these problems for so-called path–treebipartite graphs, which are a subclass of bipartite graphs between tree convex bipartite graphs and convex bipartite graphs. A bipartite graph G with bipartition (X, Y) is called a path-tree bipartite graph, if a tree T that is defined on X exists such that, for all vertices y in Y, the neighbors of y form a path in T. This study reveals that the problems of counting independent sets and maximal independent sets remain #P-complete even for path-tree bipartite graphs but a stricter restriction to rooted path–tree bipartite graphs admits polynomial-time solutions.