Some Families of Graphs with Small Power Domination Number

TL;DR

This paper investigates specific families of graphs where the power domination number is either 1 or 2, providing insights into their structural properties and minimal dominating sets.

Contribution

The paper introduces new families of graphs with small power domination numbers, expanding understanding of domination parameters in graph theory.

Findings

Identified graph families with power domination number 1

Identified graph families with power domination number 2

Provided structural characterizations of these families

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.