Graphs with disjoint 2-dominating sets

TL;DR

This paper characterizes minimal graphs that contain two disjoint 2-dominating sets, expanding understanding of domination properties in graph theory.

Contribution

It provides two new characterizations of minimal (2,2)-dominated graphs, a class defined by disjoint 2-dominating sets.

Findings

Two characterizations of minimal (2,2)-dominated graphs

Enhanced understanding of domination in graph theory

Framework for identifying such graphs

Abstract

A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G\setminus D$ has a neighbor in $D$, while $D$ is a 2-dominating set of $G$ if every vertex belonging to $V_G\setminus D$ is joined by at least two edges with a vertex or vertices in $D$. A graph $G$ is a $(2,2)$-dominated graph if it has a pair $(D,D')$ of disjoint $2$-dominating sets of vertices of $G$. In this paper we present two characterizations of minimal $(2,2)$-dominated graphs.