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 \( \beta \)-\emph{packing number} of \( G \), denoted \( \beta\text{-pack}(G) \), equals the maximum cardinality of a \( \beta \)-packing set in \( G \).
In this research, we determine \( \beta\text{-pack}(G) \) for several classes of graphs, and we explore some properties of \( \beta \)-packing sets.
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