Advanced Domination Concepts In Product Bipolar Fuzzy Graphs: Theory, Operations, And Applications

Abstract

Bipolar fuzzy graphs (BFGs) extend classical fuzzy graph theory by in-corporating both positive and negative membership degrees, enabling the representation of dual-aspect uncertainty in complex systems. Product bipo-lar fuzzy graphs (PBfGs) provide a refined framework for modeling interde-pendent relationships where edge strength is determined muftipficatirefy by vertex attributes, rather than by the classical minimum or maximum opera-tors. This paper formulates and investigates four key advanced domination variants within the strict product-based constraints of PBfGs: secure dom-ination, which ensures network resilience under node failure; 2-domination, which guarantees redundant coverage for fault tolerance; connected perfect domination, which enforces structural connectivity and exactness of cover-age; and tiiùofe edge domination, which extends control mechanisms to edge-based network architectures. We establish theoretical foundations, prove characterization theorems and bounds, examine operational properties un-der graph products (Cartesian product, composition, union, and join), and demonstrate a real-world application to secure transit network design.