Convex-Fair Domination in Graphs

Abstract

In this paper, we propose and investigate a novel domination parameter in graph theory, termed convex-fair domination, which integrates the principles of convex and fair domination. The newly introduced concept aims to balance the structural coherence of convex dominating sets with the equitable vertex coverage characteristic of fair domination.We develop a formal definition of convex-fair domination and present a comprehensive theoretical framework to support its analysis. Several theorems, lemmas, and corollaries are established to explore the fundamental properties, bounds, and existence conditions of convex-fair dominating sets. Through illustrative examples and proofs, we demonstrate the applicability of this parameter across various graph classes. Additionally, the paper encourages theexamination of convex-fair domination in new and complex graph families, laying the groundwork for future research in this direction. The introduction of this parameter not only enriches the current landscape of domination theory but also opens pathways for its application in network optimization, resource allocation, and equitable system design.