Domination Value in \(P_{2}\Box P_{n}\) and \(P_{2}\Box C_{n}\)

Abstract

A set \( D \subseteq V(G) \) is a dominating set of a graph \( G \) if every vertex of \( G \) not in \( D \) is adjacent to at least one vertex in \( D \). A minimum dominating set of \( G \), also called a \( \gamma(G) \)-set, is a dominating set of \( G \) of minimum cardinality. For each vertex \( v \in V(G) \), we define the domination value of \( v \) to be the number of \( \gamma(G) \)-sets to which \( v \) belongs. In this paper, we find the total number of minimum dominating sets and characterize the domination values for \( P_2 \Box P_n \), and \( P_2 \Box C_n \).