Integrity and domination integrity of gear graphs

Abstract

C.A. Barefoot, et. al. [4] introduced the concept of the integrity of a graph. It is an useful measure of vulnerability and it is defined as follows. I(G) = min{|S| + m(G ? S) : S ? V (G)}, where m(G ? S) denotes the order of the largest component in G ? S. Unlike the connectivity measures, integrity shows not only the difficulty to break down the network but also the damage that has been caused. A subset S of V (G) is said to be an I-set if I(G) = |S| + m(G ? S). We introduced a new vulnerability parameter in [4],namely domination integrity of a graph G. It is a defined as DI(G) = min{|S| + m(G ? S)}, where S is a dominating set of G and m(G ? S) denotes the order of the largest component in G ? S. K.S. Bagga,et. al. [2] gave a formula for I(K2 × Cn). In this paper, we give a correct formula for I(K2 × Cn). We find some results on the integrity and domination integrity of gear graphs.