Vertex-neighbor-integrity of magnifiers, expanders, and hypercubes

AbstractA set of vertices S is subverted from a graph G by removing the closed neighborhood N[S] from G. We denote the survival subgraph of the vertex subversion strategy S by G/S. The vertex-neighbor-integrity of G is defined to be VNI(G)=minS⊆V(G){|S|+ω(G/S)}, where ω(H) is the order of the largest connected component in the graph H. The graph parameter VNI was introduced by Cozzens and Wu [3] to measure the vulnerability of a spy network. Cozzens and Wu showed that the VNI of paths, cycles, trees and powers of paths on n vertices are all on the order of n. Here we prove that the VNI of any member of a family of magnifier graphs is linear in the order of the graph. We also find upper and lower bounds on the VNI of hypercubes. Finally, we show that the decision problem corresponding to computing the vertex-neighbor-integrity of a graph is NP-complete