Strong coalitions in graphs

TL;DR

This paper introduces the concept of strong coalitions in graphs, exploring their properties and defining the strong coalition number as a new graph invariant related to dominating sets.

Contribution

It defines the novel concept of strong coalitions and the strong coalition number, expanding the understanding of dominating set structures in graph theory.

Findings

Defined strong coalition and strong coalition partition concepts

Established properties and bounds of the strong coalition number

Analyzed the structure of strong coalitions in various graph classes

Abstract

For a graph $G=(V,E)$, a set $D\subset V(G)$ is a strong dominating set of $G$, if for every vertex $x\in V (G)\setminus D$ there is a vertex $y\in D$ with $xy \in E(G)$ and $deg(x)\leq deg(y)$. A strong coalition consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a strong dominating set but whose union $V_{1}\cup V_{2}$, is a strong dominating set. A vertex partition $\Omega=\{V_1, V_2,..., V_k \}$ of vertices in $G$ is a strong coalition partition, if every set $V_i \in\Omega$ either is a strong dominating set consisting of a single vertex of degree $n-1$, or is not a strong dominating set but produces a strong coalition with another set $V_j \in \Omega$ that is not a strong dominating set. The maximum cardinality of a strong coalition partition of $G$ is the strong coalition number of $G$ and is denoted by $SC(G)$. In this paper, we study properties of strong coalitions in graphs.