Perfect set domination in neutrosophic graphs

Abstract

Domination in graphs has been extensively studied in classical settings; however, its extension to uncertainty-based frameworks remains a growing area of interest. In this paper, we introduce the notion of perfect set domination in neutrosophic graphs-a framework that generalizes classical graph theory by assigning truth, indeterminacy, and falsity membership values to both vertices and edges. We formally define a perfect set dominating set (PSDS) within this neutrosophic setting and develop a comprehensive theory encompassing structural theorems, sharp bounds, and characterization results. Specifically, we prove that every neutrosophic graph can be embedded in a larger neutrosophic graph that admits a distance-d perfect set dominating set. We derive a sharp lower bound on the perfect set domination number in terms of the order and maximum degree of the graph, and characterize perfect set dominating sets via maximal independent sets. Furthermore, we investigate perfect set domination in neutrosophic trees and cube-connected cycles graphs, establishing dedicated results for each family. All theoretical findings are supported by worked examples to enhance clarity and accessibility.