Bounds on the 2-domination number

TL;DR

This paper introduces a new method for constructing 2-dominating sets in graphs, providing improved upper bounds on the 2-domination number for graphs with minimum degree between 6 and 21, showing it is less than half the number of vertices for degree at least 6.

Contribution

It presents a novel construction technique for 2-dominating sets that yields tighter bounds on the 2-domination number based on minimum degree, improving previous results.

Findings

Upper bounds on 2-domination number for δ(G) between 6 and 21

Proof that γ₂(G) < n/2 for δ(G) ≥ 6

Method using weight assignment to vertices during construction

Abstract

In a graph $G$, a set $D\subseteq V(G)$ is called 2-dominating set if each vertex not in $D$ has at least two neighbors in $D$. The 2-domination number $\gamma_2(G)$ is the minimum cardinality of such a set $D$. We give a method for the construction of 2-dominating sets, which also yields upper bounds on the 2-domination number in terms of the number of vertices, if the minimum degree $\delta(G)$ is fixed. These improve the best earlier bounds for any $6 \le \delta(G) \le 21$. In particular, we prove that $\gamma_2(G)$ is strictly smaller than $n/2$, if $\delta(G) \ge 6$. Our proof technique uses a weight-assignment to the vertices where the weights are changed during the procedure.