On the coalition number of trees

TL;DR

This paper investigates the coalition number in graphs, characterizes specific graphs and trees based on their coalition partitions, and explores the structure of coalition graphs, solving several open problems in the field.

Contribution

It characterizes graphs with maximum coalition number and trees with near-maximum coalition number, and analyzes the structure of coalition graphs for paths, addressing open problems.

Findings

Graphs with minimum degree ≤ 1 and maximum coalition number equal to the number of vertices are characterized.

All trees with coalition number one less than the number of vertices are characterized.

The number of coalition graphs for a path is determined, and it is shown that no universal coalition path exists.

Abstract

Let $G$ be a graph with vertex set $V$ and of order $n = |V|$, and let $\delta(G)$ and $\Delta(G)$ be the minimum and maximum degree of $G$, respectively. Two disjoint sets $V_1, V_2 \subseteq V$ form a coalition in $G$ if none of them is a dominating set of $G$ but their union $V_1\cup V_2$ is. A vertex partition $\Psi=\{V_1,\ldots, V_k\}$ of $V$ is a coalition partition of $G$ if every set $V_i\in \Psi$ is either a dominating set of $G$ with the cardinality $|V_i|=1$, or is not a dominating set but for some $V_j\in \Psi$, $V_i$ and $V_j$ form a coalition. The maximum cardinality of a coalition partition of $G$ is the coalition number $\mathcal{C}(G)$ of $G$. Given a coalition partition $\Psi = \{V_1, \ldots, V_k\}$ of $G$, a coalition graph $\CG(G, \Psi)$ is associated on $\Psi$ such that there is a one-to-one correspondence between its vertices and the members of $\Psi$, where two vertices of $\CG(G, \Psi)$ are adjacent if and only if the corresponding sets form a coalition in $G$. In this paper, we partially solve one of the open problems posed in Haynes et al. \cite{coal0} and we solve two open problems posed by Haynes et al. \cite{coal1}. We characterize all graphs $G$ with $\delta(G) \le 1$ and $\mathcal{C}(G)=n$, and we characterize all trees $T$ with $\mathcal{C}(T)=n-1$. We determine the number of coalition graphs that can be defined by all coalition partitions of a given path. Furthermore, we show that there is no universal coalition path, a path whose coalition partitions defines all possible coalition graphs.