2-power domination number for Knodel graphs and its application in communication networks

In a graph G, if each node v∈V (G)\S is connected to some node in S, then the set S of nodes is referred to as a dominating set. The domination number of G is the minimum cardinality of all dominating sets of G and is represented by γ(G). If a dominating set S monitors every node in the system under a set of guidelines for power systems monitoring, then the set S is referred to as a power-dominating set of G. The power domination number of G is the least number of vertices of a power dominating set of G. A generalization of power domination is the k-power domination in a graph G. The k-power domination number of G is the minimum cardinality of all k-power dominating sets of G and is represented by γp,k(G). In this paper, we have obtained the 2-power domination number represented by γp,2(G) for 4-regular Kn¨odel graphs and given the lower bound for 5-regular Kn¨odel graphs.