Resilient Graph Identification Models Based on Identifying Secure Domination

Abstract

In this research paper we investigates the identifying secure domination number, a parameter that merges two fundamental ideas in graph theory: the uniqueness property of identifying codes and the resilience property of secure domination. This hybrid parameter ensures that each vertex of a connected graph is simultaneously recognized by a unique neighborhood pattern and remains defended under vertex replacement operations.

We establish new structural bounds, derive sharp formulas for several product graphs, and introduce transfer principles that relate the identifying secure domination number of product graphs to those of their constituent factors. Our analysis covers Cartesian and lexicographic products and extends classical results for paths to broader graph families such as trees, chordal graphs, graphs of bounded degree, and twin-free classes. Several new theorems presented here highlight the interplay between adjacency patterns in product layers and the mechanisms of identification and secure domination. The results developed in this paper provide a deeper understanding of hybrid domination parameters and open new directions for research in product graph theory.