Domination Value in $P_2\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{P}_n$ and $P_2\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{C}_n$

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 $P_2\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{P}_n$, and $P_2\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{C}_n$.