Vertex Contraction Operations in Advanced Fuzzy Graph Types: Intuitionistic, Pythagorean, Neutrosophic, and Bipolar Extensions

Abstract

This paper extends the fundamental concept of vertex contraction in fuzzy graphs to advanced fuzzy graph structures, including intuitionistic fuzzy graphs (IFGs), Pythagorean fuzzy graphs (PFGs), neutrosophic graphs (NGs), and bipolar fuzzy graphs (BFGs). Formal definitions of contraction operations are developed to preserve the essential characteristics of each graph type. The effects of these operations on domination parameters are examined, and several theoretical properties are established, including the preservation of hesitancy degrees, the behavior of membership functions, and structural invariants. The proposed framework offers a unified approach to graph simplification and network analysis in uncertain environments characterized by multiple dimensions of uncertainty. Applications to multi-criteria decision-making, social network analysis, and conflict resolution systems are also discussed.