$\beta$-Packing Sets in Graphs

A set $S\subseteq V$ is $\alpha$-dominating if for all $v\in V-S$, $|N(v) \cap S | \geq \alpha |N(v)|.$ The $\alpha$-domination number of $G$ equals the minimum cardinality of an $\alpha$-dominating set $S$ in $G$. Since being introduced by Dunbar, et al. in 2000, $\alpha$-domination has been studied for various graphs and a variety of bounds have been developed. In this paper, we propose a new parameter derived by flipping the inequality in the definition of $\alpha$-domination. We say a set $S \subset V$ is a $\beta$-packing set of a graph $G$ if $S$ is a proper, maximal set having the property that for all vertices $v \in V-S$, $|N(v) \cap S| \leq \beta |N(v)|$ for some $0 < \beta \leq 1.$ The $\beta$-packing number of $G$ ($\beta$-pack($G$)) equals the maximum cardinality of a $\beta$-packing set in $G$. In this research, we determine $\beta$-pack($G$) for several classes of graphs, and we explore some properties of $\beta$-packing sets. Keywords: $\beta$-packing, $\alpha$-domination, graph theory, graph parameters