A note on $I$-sets in graphs

R. Sundareswaran, V. Swaminathan

C.A. Barefoot, et. al. introduced the concept of the integrity of a graph. It is an useful measure of vulnerability and it is defined as follows. $I(G)=\min\{|G|+m(G-S):S\subset V(G)$ where $m(G-S)$ denotes the order of the largest component in $G-S\}$. Unlike the connectivity measures, integrity shows not only the di±culty 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 $I(G)=|S|+m(G-S)$. In this paper, we de¯ne the $I$-critical graphs, $I$-excellent graphs and Bondage Integrity number and we study these parameters.