A Note on Edge Coalitions in Graphs

TL;DR

This paper extends the concept of coalitions in graphs from vertices to edges, introducing edge coalitions and analyzing their existence, structure, and properties in various graph classes and trees.

Contribution

It introduces the novel concept of edge coalitions in graphs, expanding the vertex-based coalition theory to an edge-based perspective and characterizing their structures.

Findings

Edge coalitions exist in specific graphs and trees.

Graphs with few edge coalitions are characterized.

Edge coalition structures are analyzed in special graph classes.

Abstract

Haynes et al. (2020) introduced and investigated the concept of coalition in graphs \cite{hhhmm1}. Their study examined this concept from a vertex-based perspective, whereas in this paper, we extend the investigation to an edge-based perspective of graphs. \\ An edge coalition in a graph $G=(V,E)$ consists of two disjoint sets of edges $E_1$ and $E_2$, neither of which individually forms an edge dominating set, but whose union $E_1\cup E_2$ is an edge dominating set. An edge coalition partition in a graph $G$ of order $n=|V|$ and size $|E|=m$ is an edge partition $\pi=\{E_1,\cdots,E_k\}$ so that every set $E_i$ of $\pi$ either is a singleton edge dominating set, or is not an edge dominating set but forms an edge coalition with another set $E_j$ in $\pi$, which is also not an edge dominating set. In this paper, we introduce the concept of an edge coalition and demonstrate its existence in particular graphs and trees. Additionally, we characterize graphs with small number of edge coalitions and analyze edge coalition structures in various special graph classes.