Secure coalitions in graphs

TL;DR

This paper introduces the concept of secure coalitions in graphs, explores their properties, and characterizes graphs based on their secure coalition number, including proofs that all graphs admit such partitions.

Contribution

It defines secure coalition partitions, introduces secure coalition graphs, and characterizes graphs with specific secure coalition numbers, including all trees and graphs without isolated vertices.

Findings

Every graph admits a secure coalition partition.

Characterization of graphs with SEC(G) in {1, 2, n}.

All trees have SEC(T) = n-1.

Abstract

A secure coalition in a graph $G$ consists of two disjoint vertex sets $V_1$ and $V_2$, neither of which is a secure dominating set, but whose union $V_1 \cup V_2$ forms a secure dominating set. A secure coalition partition ($sec$-partition) of $G$ is a vertex partition $\pi = \{V_1, V_2, \dots, V_k\}$ where each set $V_i$ is either a secure dominating set consisting of a single vertex of degree $n-1$, or a set that is not a secure dominating set but forms a secure coalition with some other set $V_j \in \pi$. The maximum cardinality of a secure coalition partition of $G$ is called the secure coalition number of $G$, denoted $SEC(G)$. For every $sec$-partition $\pi$ of a graph $G$, we associate a graph called the secure coalition graph of $G$ with respect to $\pi$, denoted $SCG(G,\pi)$, where the vertices of $SCG(G,\pi)$ correspond to the sets $V_1, V_2, \dots, V_k$ of $\pi$, and two vertices are adjacent in $SCG(G,\pi)$ if and only if their corresponding sets in $\pi$ form a secure coalition in $G$. In this study, we prove that every graph admits a $sec$-partition. Further, we characterize the graphs $G$ with $SEC(G) \in \{1,2,n\}$ and all trees $T$ with $SEC(T) = n-1$. Finally, we show that every graph $G$ without isolated vertices is a secure coalition graph.