Graph Operations and Neighbor Rupture Degree

Abstract

In a communication network, the vulnerability parameters measure the resistance of the network to disruption of operation after the failure of certain stations or communication links. A vertex subversion strategy of a graph $G$, say $S$, is a set of vertices in $G$ whose closed neighborhood is removed from $G$. The survival subgraph is denoted by $G/S$. The neighbor rupture degree of $G$, $\text{N}\text{r}(G)$, is defined to be $\text{N}\text{r} (G) =\text{max}\{w(G/S) -|S| - c(G/S): S \subset V(G), w(G/S) \ge 1\}$, where $S$ is any vertex subversion strategy of $G$, $w(G/S)$ is the number of connected components in $G/S$ and $c(G/S)$ is the maximum order of the components of $G/S$ (G. Bacak Turan, 2010). In this paper we give some results for the neighbor rupture degree of the graphs obtained by some graph operations.