Extending Hesitant Fuzzy Graphs to Bipolar Hesitant Frameworks for Capturing Positive and Negative Uncertainty

Abstract

Hesitant fuzzy graphs provide a flexible framework for representing uncertainty through multiple possible membership values assigned to vertices and edges. However, classical hesitant fuzzy graphs are restricted to positive hesitant information and do not incorporate negative or conflicting influence. In this paper, we introduce a Bipolar Hesitant Fuzzy Graph (BHFG) framework that extends hesitant fuzzy graphs by simultaneously incorporating positive and negative hesitant membership degrees. Formal definitions and consistency conditions are established, and it is proved that every hesitant fuzzy graph can be embedded in the bipolar hesitant framework, while the converse does not hold. Fundamental structural operations, including complement, union, and Cartesian product, are investigated and shown to preserve the bipolar hesitant structure. A bipolar hesitant uncertainty measure is proposed to quantify combined positive and negative hesitation. An illustrative application to a conflicting social relationship network demonstrates the enhanced expressive capability of the proposed framework compared to classical hesitant fuzzy graphs. The results confirm that the bipolar hesitant model provides a more comprehensive representation of networks involving cooperative and conflicting interactions.