Properties of Partial Dominating Sets of Graphs

Abstract

A set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V–S$ is adjacent to at least one vertex in $S$. The domination number $\gamma (G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a set $S$ is a $\gamma$-set.

A generalization of this is partial domination, which was introduced in 2017 by Case, Hedetniemi, Laskar, and Lipman. In \emph{partial domination}, a set $S$ is a $p$-dominating set if it dominates a proportion $p$ of the vertices in $V$. The $p$-domination number ${\gamma }_p(G)$ is the minimum cardinality of a $p$-dominating set in $G$.

In this paper, we investigate further properties of partial dominating sets, particularly ones related to graph products and locating partial dominating sets. We also introduce the concept of a $p$-influencing set as the union of all $p$-dominating sets for a fixed $p$ and investigate some of its properties.