Maximal independent sets in bipartite graphs with at least one cycle

A maximal independent set is an independent set that is not a proper subset of any other independent set. Liu [J.Q. Liu, Maximal independent sets of bipartite graphs, J. Graph Theory, 17 (4) (1993) 495-507] determined the largest number of maximal independent sets among all n-vertex bipartite graphs. The corresponding extremal graphs are forests. It is natural and interesting for us to consider this problem on bipartite graphs with cycles. Let Bn (resp. B′n) be the set of all n-vertex bipartite graphs with at least one cycle for even (resp. odd) n. In this paper, the largest number of maximal independent sets of graphs in Bn (resp. B′n) is considered. Among Bn the disconnected graphs with the first-, second-,..., n−2 2-th largest number of maximal independent sets are characterized, while the connected graphs in Bn having the largest and the second largest number of maximal independent sets are determined. Among B n graphs having the largest number of maximal independent sets are identified