Domination and 2-degree-packing numbers in graphs

TL;DR

This paper establishes a bound relating the domination number and the 2-degree-packing number in graphs, proving that the domination number is at most one less than the 2-degree-packing number, and characterizes graphs where equality holds.

Contribution

It proves a new inequality between domination and 2-degree-packing numbers and characterizes the graphs where the bound is tight.

Findings

Proves that $orall$ simple graph $G$, $ ext{domination number} \\leq ext{2-degree-packing number} - 1$.

Provides a characterization of connected graphs satisfying $\\gamma(G) = u_2(G) - 1$.

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 dominating set. A subset $R$ of edges of a graph $G$ is a 2-degree-packing, if any three edges from $R$ do not have the same incident vertex. The 2-degree-packing number of $G$, $\nu_2(G)$, is the maximum order of a 2-degree-packing of $G$. In this paper, we prove that any simple graph $G$ satisfies $\gamma(G)\leq\nu_2(G)-1$. Furthermore, we give a characterization of simple connected graphs $G$ satisfying $\gamma(G)=\nu_2(G)-1$.