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

The Average Lower Connectivity of Graphs

For a vertex v of a graph G, the lower connectivity, denoted by sv(G), is the smallest number of vertices that contains v and those vertices whose deletion from G produces a disconnected or a trivial graph. The average lower connectivity denoted by κav(G) is the value (∑v∈VGsvG)/VG. It is shown that this parameter can be used to measure the vulnerability of networks. This paper contains results on bounds for the average lower connectivity and obtains the average lower connectivity of some graphs