A Vulnerability Parameter of Networks

Abstract

The vulnerability in a communication network is the measurement of the strength of the network against damage that occurs in nodes or communication links. It is important that a communication network is still effective even when it loses some of its nodes or links. In other words, since a network can be modelled by a graph, it is desired to know whether the graph is still connected when some of the vertices or edges are removed from a connected graph. The vulnerability parameters aim to find the nature of the network when a subset of the nodes or links is removed. One of these parameters is domination. Domination is a measure of the connection of a subset of vertices with its complement. In this paper, we study porous exponential domination as a vulnerability parameter and obtain certain results on the Cartesian product and lexicographic product graphs. We determine the porous exponential domination number, denoted by. gamma*(e), of the Cartesian product of P-2 with P-n and C-n, separately. We also determine the porous exponential domination number of the Cartesian product of P-n with complete bipartite graphs and any graph G which has a vertex of degree vertical bar V(G)vertical bar - 1. Moreover, we obtain the porous exponential domination number of the lexicographic product of P-n and G(m), denoted by P-n[G(m)], for the case where G(m) is a graph of order m with a vertex of degree m - 1 and for the opposite case where G(m) is a graph of order m which has no vertex of degree m - 1. We further show that gamma*(e)(P-n[G(m)]) = gamma*(e)(G(m)[P-n]) = gamma*(e)(P-n) by proving gamma*(e)(G(m)[G(n)]) = gamma*(e)(G(n)), where G(m) is a graph of order m with a vertex of degree m - 1 and G(n) is any graph of order n.