Some Families of Graphs with Small Power Domination Number

Abstract

Let $ G $ be a graph with the vertex set $ V(G) $ and $ S $ be a subset of $ V(G) $. Let $cl(S)$ be the set of vertices built from $S$, by iteratively applying the following propagation rule: if a vertex and all of its neighbors except one of them are in $ cl(S) $, then the exceptional neighbor is also in $ cl(S) $. A set $S$ is called a zero forcing set of $G$ if $cl(S)=V(G)$. The zero forcing number $Z(G)$ of $G$ is the minimum cardinality of a zero forcing set. Let $cl(N[S])$ be the set of vertices built from the closed neighborhood $N[S]$ of $S$, by iteratively applying the previous propagation rule. A set $S$ is called a power dominating set of $G$ if $cl(N[S])=V(G)$. The power domination number $\gamma_p (G)$ of $G$ is the minimum cardinality of a power dominating set. In this paper, we present some families of graphs that their power domination number is 1 or 2.