The Role of Domination and Sign Domination Numbers in Network Stability: A Graph-Theoretic Approach

• Author(s): M. S. Patil
• Paper ID: 1702982
• Page: 328-337
• Published Date: 30-11-2021
• Published In: Iconic Research And Engineering Journals
• Publisher: IRE Journals
• e-ISSN: 2456-8880
• Volume/Issue: Volume 5 Issue 5 November-2021

Abstract

This research explores the intricate relationship between domination numbers and sign domination numbers in the context of network stability through a graph-theoretic lens, focusing on the conceptual interplay of these parameters as critical measures of influence, resilience, and resource optimization in network structures, where the domination number—a fundamental parameter in graph theory—quantifies the minimum number of vertices required to dominate all other vertices in a graph, while the sign domination number extends this concept by incorporating signed functions to refine the notion of domination in graphs with weighted or directed edges, and investigates how these parameters influence the robustness and fault-tolerance of networks, particularly in systems modeled as social, biological, or communication networks, highlighting that lower domination numbers often signify optimal resource allocation and control in static graphs while higher sign domination numbers can indicate greater flexibility in adapting to dynamic environments, with theoretical frameworks analyzing variations in these metrics across graph classes such as bipartite graphs, planar graphs, and dense graphs, and presenting new bounds, inequalities, and characterizations of domination and sign domination numbers in diverse topological settings; furthermore, the study delves into the role of graph transformations including edge additions, vertex removals, and subgraph operations—on the stability of these parameters, revealing how structural perturbations impact network resilience, particularly in minimizing vulnerabilities caused by cascading failures or targeted attacks, and demonstrates through theoretical modeling how sign domination numbers, with their nuanced approach to assigning signed functions, provide a more flexible and adaptive framework for evaluating network stability in weighted and dynamic graphs compared to traditional domination numbers, offering insights into optimizing control, coverage, and influence in practical applications such as communication protocols, epidemic containment strategies, and sensor network designs, while also presenting open problems and conjectures for future research in extending the comparative analysis of domination and sign domination numbers to hypergraphs, multi-layered graphs, and time-evolving networks, ultimately contributing to a deeper theoretical understanding of how these graph-theoretic parameters govern the stability, adaptability, and efficiency of complex network systems under varying structural and functional constraints, thereby paving the way for innovative solutions in the design and analysis of resilient and efficient networks.

Keywords

Domination Number, Sign Domination Number, Network Stability, Graph Theory, Network Resilience, Graph Transformations

Citations

IRE Journals:
M. S. Patil "The Role of Domination and Sign Domination Numbers in Network Stability: A Graph-Theoretic Approach" Iconic Research And Engineering Journals Volume 5 Issue 5 2021 Page 328-337

IEEE:
M. S. Patil "The Role of Domination and Sign Domination Numbers in Network Stability: A Graph-Theoretic Approach" Iconic Research And Engineering Journals, 5(5)