On Coalition Graphs and Coalition Count of Graphs

TL;DR

This paper explores the concept of coalition graphs and counts in graphs, constructing smaller graphs with specific coalition properties and characterizing graphs with a coalition count of one.

Contribution

It constructs smaller graphs with given coalition graphs and characterizes all graphs with a coalition count of one, advancing understanding of coalition structures in graphs.

Findings

Constructed smaller graphs with specified coalition graphs.

Characterized all graphs with coalition count equal to one.

Studied properties of coalition count under certain conditions.

Abstract

Let $G$ be graph with vertex set $V(G)$ and order $n$. A coalition in a graph $G$ consists of two disjoint sets of vertices $V_1$ and $V_2$, neither of which is a dominating set but whose union $V_1 \cup V_2$ is a dominating set. A coalition partition, abbreviated $c$-partition, in a graph $G$ is a vertex partition $\pi=\left\{V_1 , V_2,\dots, V_k\right\}$ such that every set $V_i$ of $\pi$ is either a singleton dominating set, or is not a dominating set but forms a coalition with another set $V_j$ in $\pi$. The sets $V_i$ and $V_j$ are coalition partners in $G$. The coalition number $C(G)$ equals the maximum order $k$ of a $c$-partition of $G$. For any graph $G$ with a $c$-partition $\pi=\left\{V_1,V_2,\dots,V_k\right\}$, the coalition graph $CG(G,\pi)$ of $G$ is a graph with vertex set $V_1,V_2,\dots, V_k$, corresponding one-to-one with the set $\pi$, and two vertices $V_i$ and $V_j$ are adjacent in $CG(G,\pi)$ if and only if the sets $V_i$ and $V_j$ are coalition partners in $\pi$. In [4], authors proved that for every graph $G$ there exist a graph $H$ and $c$-partition $\pi$ such that $CG(H,\pi)\cong G$, and raised the question: Does there exist a graph $H^*$ of smaller order $n^*$ and size $m^*$ with a $c$-partition $\pi^*$ such that $CG(H^*,\pi^*)\cong G$?. In this paper, we constructed a graph $H^*$ of small order and size and a $c$- partition $\pi^*$ such that $CG(H^*,\pi^*)\cong G$. Recently, Haynes et al.[5] defined the coalition count $c(G)$ of a graph $G$ as the maximum number of different coalition in any $c$-partition of $G$. We characterize all graphs $G$ with $c(G)=1$. Further, imposing some suitable conditions on coalition number, we study the properties of coalition count of graph.