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 $n$.