Roman domination parameters with respect to differentials in probabilistic neural networks

The differential theory of graphs is perfectly integrated into domination theory, providing a robust framework for investigating Roman domination parameters without relying on functions. By applying proven Gallai-type theorems that link differentials and Roman domination parameters, this study highlights the practical advantages of computing either parameter. Specifically, this paper focuses on computing the differential, 2-packing differential, perfect differential, and restrained differential of three- and four-layered probabilistic neural networks (PNNs). Given that evaluating such network invariants is critical to understanding structural dynamics, we address relevant NP-complete problems by analyzing the topological graph configurations of these PNNs. Furthermore, a comprehensive structural characterization of the dominant differential and Roman graph classes within these network architectures is provided.