Global coalition sets in graphs

Abstract

Let $G=(V,E)$ be a graph. A subset $S\subseteq V$ is called a global dominating set of $G$, if it serves as a dominating set in both $G$ and its complement $\overline{G}$. We define two disjoint subsets $V_1,V_2\subseteq V$ to form a global coalition if neither $V_1$ nor $V_2$ individually constitutes a global dominating set, yet their union $V_1\cup V_2$ does. A global coalition partition (abbreviated as $gc$-partition) of $G$ is a vertex partition $\pi$ of $V(G)$ such that for every subset $V_i\in \pi$, there exists another subset $V_j\in \pi$ with which $V_i$ forms a global coalition. In this paper, we initiate the study of global coalition in graphs. Specifically, we prove that every graph admits a gc-partition. Additionally, we establish an upper bound on the number of global coalitions in which each member of a gc-partition can participate. We also explore the relationships between global coalition and coalition, as well as between global coalition and perfect coalition in graphs. Finally, we explore properties of $gc$-partitions in unicyclic graphs.

Unfortunately, the second version of the article was uploaded to arXiv hastily and without consulting all of the authors, and it contains some incorrect results. Furthermore, revising the paper and preparing a new version will take some time