Vertex-neighbour-integrity of composition graphs of paths and cycles

A vertex subversion strategy of a graph G is a set of vertices X∪ V(G) whose closed neighbourhood is deleted from G. The survival subgraph is denoted by G/X. The vertex-neighbour-integrity of G is defined to be VNI(G)=min{|X|+τ(G/X):X∪ V(G)}, where τ(G/X) is the maximum order of the components of G/X. This graph parameter was introduced by Cozzens and Wu to measure the vulnerability of spy networks. Gambrell proved that the decision problem of computing the vertex-neighbour-integrity of a graph is NP-complete. In this paper we evaluate the vertex-neighbour-integrity of the composition graphs of paths and cycles.