GLOBAL FAIR DOMINATION NUMBER OF SOME GRAPHS

Abstract

Let G=(V,E) be a finite, simple, and undirected graph. In this paper, we present a new concept in graph theory known as the global fair domination number. A fair dominating set S in a graph G=(V,E) is called a global fair dominating set if it ¯ also functions as a fair dominating set in the complement graph G . The smallest cardinality of such a set is referred to as the global fair domination number of G, denoted by γgfd(G). In this study, we investigate global fair dominating sets and determine the global fair domination number for various classes of graphs, including path graphs, cycle graphs, wheel graphs, star graphs, ladder graphs, k-partite graphs, and prism graphs. The obtained results reveal recurring structural patterns that facilitate both efficient and balanced coverage in graph-based systems. From an application perspective, these results provide a mathematical framework for modeling medical networks, where healthcare facilities and their communication or referral links can be represented as vertices and edges, respectively. The fairness condition promotes equitable healthcare service coverage, while the global requirement enhances the robustness of the medical network under both normal and alternative connectivity conditions.