Comparative Analysis of Soft Directed Graph Products Through Soft Subdigraph Relations

Soft set theory has emerged as an effective mathematical tool for handling uncertainty and has been widely applied in decision-making problems. In graph theory, directed graphs are fundamental structures in which vertices are connected by directed edges, called arcs. A soft directed graph integrates soft set theory with directed graphs, offering a parameterized framework for studying complex relationships under uncertainty. Since graph product operations play a central role in structural graph theory and its applications, their extension to soft directed graphs is of considerable interest. In this paper, we conduct a comparative study of soft directed graph products by investigating their relationships through soft subdigraphs. We establish that the Cartesian product of two soft directed graphs is a soft subdigraph of the lexicographic, disjunctive, and normal products. The lexicographic product is further shown to be a soft subdigraph of the disjunctive product. We also prove that the categorical product is a soft subdigraph of the lexicographic, modular, normal, and disjunctive products. In addition, the normal product is identified as a soft subdigraph of both the lexicographic and disjunctive products. Finally, we demonstrate that the rooted product is a soft subdigraph of the Cartesian, lexicographic, disjunctive, and normal products. The findings present a systematic framework for comparing soft directed graph products, clarify their hierarchical structure, and demonstrate their importance in addressing problems characterized by uncertainty and complex network structures.