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 \( |V(G)|-1 \). 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 \( n \).
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Çiftçi, C., Aytaç, A. A Vulnerability Parameter of Networks. Math Notes 109, 517–526 (2021). https://doi.org/10.1134/S0001434621030202
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DOI: https://doi.org/10.1134/S0001434621030202