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

Abstract

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.

Keywords:

• Composition graph
• Cycle
• Path
• Vertex-dominating number
• Vertex-neighbour-integrity

CCS Category::

• C.2.0
• G.2.2

Acknowledgements

This work was supported by NSFC (No.60642002), SRF for ROCS of SEM, and BSF (No.AJ12046) of XAUAT. The authors are grateful to the anonymous referees for helpful comments on an earlier version of this article.