On the Weak-Integrity of Graphs

Connectivity has been used in the past to describe the stability of graphs. If two graphs have the same connectivity, then it does not distinguish between these graphs. That is, the connectivity is not a good measure of graph stability. Then we need other graph parameters to describe the stability. Suppose that two graphs have the same connectivity and the order (the number of vertices or edges) of the largest components of these graphs are not equal. Hence, we say that these graphs must be different in respect to stability and so we can define a new measure which distinguishes these graphs. In this paper, the Weak-Integrity of a graph G is introduced as a new measure of stability in this sense and it is defined as I w(G)=min S?V(G){S+m e(G-S)}, where m e(G-S) denotes the number of edges of the largest component of G-S. We give the weak-integrity of graphs obtained via various operations that are unary, such as powers, and binary, such as union, composition, product and corona. © 2003 Kluwer Academic Publishers.