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 B-n (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 B-n (resp. B-n) considered. Among B n the disconnected graphs with the first-, second-,..., n-2/2-th largest number of maximal independent sets are characterized, while the connected graphs in B-n 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.