Convex-Fair Domination in Graphs

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 fairdomination. The newly introduced concept aims to balancethe structural coherence of convex dominating sets with theequitable vertex coverage characteristic of fair domination.We develop a formal definition of convex-fair dominationand 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, wedemonstrate the applicability of this parameter across various graph classes. Additionally, the paper encourages theexamination of convex-fair domination in new and complexgraph families, laying the groundwork for future research inthis direction. The introduction of this parameter not onlyenriches the current landscape of domination theory but alsoopens pathways for its application in network optimization,resource allocation, and equitable system design.