A Comprehensive Review On Vertex Dominations In Graph Theory: Exploring Theory And Applications

Abstract

This comprehensive review delves into the intricate realm of VERTEX dominations in Graph Theory, providing an extensive exploration of both theory and practical applications. The article encompasses a thorough examination of various domination aspects in graphs, including Domination in Planar graphs, connected graph dominations, edge dominations in Paths, Cycles of related graphs, and associated properties. Additionally, the study extends to inverse dominations on graphs, shedding light on their significance in real-world scenarios. In graph theory, the idea of dominance states that a collection of vertices If every vertex in graph G is either in S or close to a vertex in S, then S dominates graph G. G's dominance number is based on the size of the least dominating set. In recent years, there has been interest in two alternative concepts: connected domination and absolute dominance. Every vertex in the graph must be next to every vertex in S for there to be a complete dominant set; nevertheless, a linked dominant set both dominates the graph and creates a connected subgraph. Numerous fields, including as radio programmes, computer communication networks, and school bus routing, may benefit from the use of these dominating concepts., social networks, and interconnection systems. The goal of the essay is to provide a comprehensive knowledge. of VERTEX dominations, establishing their theoretical foundations and illustrating their relevance in practical scenarios.