Introduction to double coalitions in graphs

TL;DR

This paper introduces the concept of double coalitions in graphs, exploring their properties and how they relate to double dominating sets and partitions, expanding understanding of graph domination structures.

Contribution

It defines and analyzes the properties of double coalitions and their partitions in graphs, a novel concept in the study of domination in graph theory.

Findings

Characterization of double coalitions in graphs

Conditions for the existence of double coalition partitions

Insights into the structure of double dominating sets

Abstract

Let $G(V, E)$ be a finite, simple, isolate-free graph. A set $D$ of vertices of a graph $G$ with the vertex set $V$ is a double dominating set of $G$, if every vertex $v\in D$ has at least one neighbor in $D$ and every vertex $v \in V \setminus D$ has at least two neighbors in $D$. A double coalition consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a double dominating set but their union $V_{1}\cup V_{2}$ is a double dominating set. A double coalition partition of a graph $G$ is a partition $\Pi = \{V_1, V_2,..., V_k \}$ of $V$ such that no subset of $\Pi$ is a double dominating set of $G$, but for every set $V_i \in \Pi$, there exists a set $V_j \in \Pi$ such that $V_i$ and $V_j$ form a double coalition. In this paper, we study properties of double coalitions in graphs.