Domination Value in $P_2 \square P_n$ and $P_2 \square C_n$

TL;DR

This paper investigates the structure and count of minimum dominating sets, as well as the domination values of vertices, in the Cartesian products of a path with two vertices and a path or cycle of length n.

Contribution

It provides explicit formulas and characterizations for the total number of minimum dominating sets and domination values in $P_2 imes P_n$ and $P_2 imes C_n$ graphs.

Findings

Derived formulas for the number of minimum dominating sets.

Characterized domination values for vertices in the studied graph classes.

Enhanced understanding of domination properties in product graphs.

Abstract

A set $D \subseteq V(G)$ is a \emph{dominating set} of a graph $G$ if every vertex of $G$ not in $D$ is adjacent to at least one vertex in $D$. A \emph{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 \emph{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 $P_2 \square P_n$ and $P_2 \square C_n$.