Vertex-neighbor-toughness of the Cartesian Products of paths and cycles

Let G = (V, E) be a graph and u is an element of V (G). The set N[u] = {u} boolean OR {v is an element of V (G)vertical bar u not equal v,u and v are adjacent} is the closed neighborhood of u. A set S subset of V (G) is called a vertex subversion strategy of G if each of the vertices in N[S] is deleted from G. By G/S we denote the survival subgraph G-N[S]. A vertex set S is called a cut strategy of G if G/S is disconnected, or is a clique, or is empty. In 2013, we introduced a parameter called vertex-neighbor-toughness to measure the neighbor invulnerability of networks [A study of network invulnerability and facility system reliability, a Dissertation for the Doctor Degree of Philosophy, Northwestern Polytechnical University, 2013, pp.55]. For a graph G, it is defined as t(VN)(G) = min{vertical bar S vertical bar/omega(G/S)}, where S is any cut strategy of G,omega(G/S) is the number of the components of G/S. In this paper, we determine the vertex-neighbor-toughness of the Cartesian Products of paths and cycles.