Relationships Between Vertex-Neighbor-Integrity and Other Parameters

Abstract

Let $G$ be a graph. A vertex subversion strategy of $G$, $S$, is a set of vertices in $G$ whose closed neighborhood is deleted from $G$. The survival-subgraph is denoted by $G/S$. The vertex-neighbor-integrity of $G$, $\mathrm{V}\mathrm{N}\mathrm{I}(G)$, is defined to be $\mathrm{V}\mathrm{N}\mathrm{I}(G)=\underset{S\subseteq V(G)}{min}\{|S|+w(G/S)\}$, where $S$ is any vertex subversion strategy of $G$, and $w(G/S)$ is the maximum order of the components of $G/S$. In this paper, we discuss the relationship between the vertex-neighbor-integrity and some well-known graphic parameters.