$\beta$Packing Sets in Graphs
Abstract
A set $S\subseteq V$ is $\alpha$dominating if for all $v\in VS$, $N(v) \cap S  \geq \alpha N(v).$ 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 $v \in VS$, $N(v) \cap S \leq \beta N(v)$ 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
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 DOI:
 10.48550/arXiv.1906.00073
 arXiv:
 arXiv:1906.00073
 Bibcode:
 2019arXiv190600073C
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 First presented at the 50th Southeastern International Conference on Combinatorics, Graph Theory &