A characterization of trees with equal 2-domination and 2-independence numbers

TL;DR

This paper offers a new constructive characterization of trees where the 2-domination number equals the 2-independence number, focusing solely on local properties during the construction process.

Contribution

It introduces a local-property-based constructive characterization of such trees, improving upon previous global-property-based methods.

Findings

Provides a local property-based characterization of trees with equal 2-domination and 2-independence numbers.

Simplifies the process of identifying these trees by focusing on local properties.

Enhances understanding of tree structures related to domination and independence parameters.

Abstract

A set $S$ of vertices in a graph $G$ is a $2$-dominating set if every vertex of $G$ not in $S$ is adjacent to at least two vertices in $S$, and $S$ is a $2$-independent set if every vertex in $S$ is adjacent to at most one vertex of $S$. The $2$-domination number $\gamma_2(G)$ is the minimum cardinality of a $2$-dominating set in $G$, and the $2$-independence number $\alpha_2(G)$ is the maximum cardinality of a $2$-independent set in $G$. Chellali and Meddah [{\it Trees with equal $2$-domination and $2$-independence numbers,} Discussiones Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive characterization of trees with equal $2$-domination and $2$-independence numbers. Their characterization is in terms of global properties of a tree, and involves properties of minimum $2$-dominating and maximum $2$-independent sets in the tree at each stage of the construction. We provide a constructive characterization that relies only on local properties of the tree at each stage of the construction.