Maximal independent sets in bipartite graphs with at least one cycle

Shuchao Li,Huihui Zhang,Xiaoyan Zhang

doi:10.46298/dmtcs.607

Abstract

Graph Theory 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 \mathscrBₙ (resp. \mathscrBₙ') 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 \mathscrBₙ (resp. \mathscrBₙ') is considered. Among \mathscrBₙ the disconnected graphs with the first-, second-, \ldots, \fracn-22-th largest number of maximal independent sets are characterized, while the connected graphs in \mathscrBₙ having the largest, the second largest number of maximal independent sets are determined. Among \mathscrBₙ' graphs have the largest number of maximal independent sets are identified.

Highlights

IntroductionJin and Li [8] determined the second largest number of maximal independent sets among all graphs of order n
Given a graph G = (VG, EG), a set I ⊆ VG is independent if there is no edge of G between any two vertices of I
The set of all maximal independent sets of a graph G is denoted by MI(G) and its cardinality by mi(G)

Summary

Introduction

Jin and Li [8] determined the second largest number of maximal independent sets among all graphs of order n. Determined the n-vertex graph having the third largest number of maximal independent sets. This paper is motivated directly from [17], in which the author completely characterized the n-vertex bipartite graphs having the largest number of maximal independent sets. For the bipartite graph of order n that contains cycles, the author in [17] determined the upper bound on the number of maximal independent sets for odd n. Characterized respectively, while among the set of all connected bipartite graphs with even order, the extremal graphs having the largest and the second largest number of maximal independent sets are identified

Preliminary

Sharp bounds and extremal graphs

Concluding remark