Vertex-Neighbor-Integrity of Trees

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 \(G\), \(\mathrm{VNI}(G)\), is defined to be \(\mathrm{VNI}(G) = \displaystyle\min_{S\subseteq V(G)} \{|S| + w(G/S)\}\), 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 show the minimum and the maximum vertex-neighbor-integrity among all trees with any fixed order, and also show that for any integer \(l\) between the extreme values there is a tree with the vertex-neighbor-integrity \(l\).