An In-depth Analysis of Domination in Graphs: Investigating the Domatic Number, Its Applications, and Computational Challenges

• Author(s): M. S. Patil
• Paper ID: 1702139
• Page: 343-353
• Published Date: 30-04-2020
• Published In: Iconic Research And Engineering Journals
• Publisher: IRE Journals
• e-ISSN: 2456-8880
• Volume/Issue: Volume 3 Issue 10 April-2020

Abstract

Graph domination is a central notion of graph theory that refers to a set of vertices such that each vertex in a graph is either part of this set or adjacent to at least one vertex in that set, and the domatic number a measure of how a graph can be partitioned into disjoint dominating sets is a fundamental metric with far-reaching implications across network design, social dynamics modeling, and optimization theory that has mostly concentrated on theoretical foundations, complexity classes, and algorithmic methods for the computation of domatic numbers in different graph classes, such as trees, planar graphs, and bipartite graphs whilst dealing with the NP-completeness of the problem for general graphs, and improving the knowledge of upper and lower bounds through structural properties with key results including exact algorithms for special cases in graph families such as chordal graphs and split graphs, heuristic methods for approximating solutions in dense and sparse graphs, and computational studies relating domatic partitions to real-life applications, like distributed systems where the trade-off between resources allocation and redundancy is crucial, extending to reliable communication networks which require efficient dominating configurations for fault tolerance and connectivity, social network analysis where dominating sets can be used to model influent groups, and finally in wireless sensor networks where the use of domatic partitions improves energy-efficient clustering and fault-tolerant node coverage, all of which keep theoretical challenges ongoing like extending domatic number concepts to weighted graphs, directed graphs, and dynamic or evolving networks showing the interplay existing between combinatorial optimization, graph coloring, and domination-based parameters, whilst recent advancements have also delved into game-theoretic approaches and probabilistic methods to estimate domatic partitions, and current open questions remain in closing the gap between theoretical notions and computational feasibility for large-scale graphs, specifically in hypergraphs and geometric graphs, which emphasizes the urgency of further conceptual and algorithmic advancements to broaden the reach and efficiency of domatic number calculations across emerging interdisciplinary areas that rely on graph-theoretical solutions.

Keywords

Domatic Number, Graph Theory, Dominating Sets, Computational Complexity, Network Optimization, Algorithmic Strategies

Citations

IRE Journals:
M. S. Patil "An In-depth Analysis of Domination in Graphs: Investigating the Domatic Number, Its Applications, and Computational Challenges" Iconic Research And Engineering Journals Volume 3 Issue 10 2020 Page 343-353

IEEE:
M. S. Patil "An In-depth Analysis of Domination in Graphs: Investigating the Domatic Number, Its Applications, and Computational Challenges" Iconic Research And Engineering Journals, 3(10)