A Vulnerability Parameter of Networks
Makaleler > Tam Makale · 2021
Yazarlar: C. Çiftçi, Aysun AYTAÇ
Mathematical Notes, Uluslararası, Wos, 2021, cilt 109, ss 517-526
DOI: 10.1134/s0001434621030202
Özet
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 $$ n $$ .