A Vulnerability Parameter of Networks

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 $ |V(G)|-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 $.