Set Domination in Neutrosophic Graphs

Abstract

Set domination is a fundamental concept in graph theory, crucial for understanding and analyzing network structures. In this paper, we extend the notion of set domination to Neutrosophic graphs, which accommodate uncertain and indeterminate information. We introduce the concept of neutrosophic set domination, which aims to identify subsets of vertices in a neutrosophic graph that either dominate the entire graph or are adjacent to dominating vertices. We define neutrosophic domination sets and investigate their properties in neutrosophic graphs. Furthermore, we explore algorithms for computing neutrosophic set domination and discuss their computational complexity. Finally, we provide numerical examples to illustrate the application of neutrosophic set domination in real-world scenarios, demonstrating its effectiveness in analyzing and optimizing uncertain networks.

Introduction

Graph theory studies the relationships between vertices and edges, with applications in mathematics, computer science, IT, biosciences, and linguistics. Fuzzy graphs, introduced by Kauffman and formalized by Yeh and Rosenfeld, generalize classical graphs by associating membership values with vertices and edges. Intuitionistic fuzzy sets (Antonassov, 1984) extend fuzzy sets by including both membership and non-membership values. Neutrosophic sets (Smarandache) further generalize this concept by incorporating three independent components—truth, indeterminacy, and falsity—allowing modeling of uncertainty, inconsistency, and imprecision. Single-valued neutrosophic sets simplify these values to the [0,1] interval for practical applications.

A neutrosophic graph is defined by neutrosophic vertex and edge sets, with cardinality measures for vertices and edges. Set domination in neutrosophic graphs studies subsets of vertices (set dominating sets, SD sets) where every vertex or subset is “dominated” by at least one vertex in the set, ensuring connectivity.

Key theoretical results include:

• Every neutrosophic connected or Hamiltonian graph has a set dominating set whose complement is also a set-dominating set.

• In neutrosophic graphs without isolated vertices, the complement of a minimal SD set is also a SD set.

• For neutrosophic graphs with independent SD sets, the graph diameter is bounded (≤ 4).

These results extend classical graph domination concepts into the neutrosophic framework, enabling the analysis of graphs under uncertainty and indeterminacy.

Conclusion

Fuzzy graph and intuitionistic graph techniques fail in situations where there is indeterminacy. The neutrosophic graph performs well in these circumstances. This study develops the concepts of set domination set in a neutrosophic graph. Here, appropriate examples are used to construct the definitions of set dominating sets, minimal SD-sets, independent set dominating sets, and set domination numbers in neutrosophic graphs. Finally, the set dominance set theorems in the neutrosophic graph are derived and Numerical examples solved in MATLAB. Future developments will expand and apply the idea of the set domination set in neutrosophic graphs to real-world problems.

References

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Copyright

Copyright © 2025 Dr. P. Solairani, N. Bavithra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.