On domination and 2-degree-packing numbers in graphs

Abstract

A dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that \-every vertex of $G$ is either in $D$ or is adjacent to a vertex in $D$. The domination number of $G$, $\gamma(G)$, is the minimum order of a domi\-nating set. The domination number has become an interesting research study on several kinds of graphs, and there are many papers related to this parameter and several variants of it. In this paper, we prove that any simple graph $G$ with no isolated vertices satisfies $\gamma(G)\leq\nu_2(G)-1$, where $\nu_2(G)$ is the maximum order of a subset $R$ of edges of $G$ such that any three edges from $R$ do not have the same incident vertex. This new parameter is called the 2-degree-packing of $G$ and it is studied in a more general context but with a different name as 2-packing number, see \cite{AvilaUtilitas}. Also, in this paper, we give a characterization of simple connected graphs $G$ satisfying $\gamma(G)=\nu_2(G)-1$.