In this paper the concept of {\em weak-integrity} is introduced as a new measure of the stability of a graph \( G \) and it is defined as Iw(G)={\minS\subset V(G)\{|S|+me(G-S)\}}, where me(G-S) denotes the number of edges of a largest component of G-S. We investigate the weak-integrity of trees and compute the weak-integrity of a binomial tree and all the trees with at most 7 vertices. We also give some results about the weak-integrity of graphs obtained from binary operations.

In this paper the concept of {\em weak-integrity} is introduced as a new measure of the stability of a graph \( G \) and it is defined as Iw(G)={\minS\subset V(G)\{|S|+me(G-S)\}}, where me(G-S) denotes the number of edges of a largest component of G-S. We investigate the weak-integrity of trees and compute the weak-integrity of a binomial tree and all the trees with at most 7 vertices. We also give some results about the weak-integrity of graphs obtained from binary operations.