An improved bound on weak independence number of a graph

Abstract

A vertex v in a graph G=(V,X) is said to be weak if d(v)?d(u) for every u adjacent to v in G. A set S ? V is said to be weak if every vertex in S is a weak vertex in G. A weak set which is independent is called a weak independent set (WIS). The weak independence number w?0(G) is the maximum cardinality of a WIS. We proved that w?0(G)? p-?. This bound is further refined in this paper and we characterize the graphs for which the new bound is attained.