Fuzzy Coalition Graphs: A Framework for Understanding Cooperative Dominance in Uncertain Networks

In a fuzzy graph (Formula presented.), a fuzzy coalition is formed by two disjoint vertex sets (Formula presented.) and (Formula presented.), neither of which is a strongly dominating set, but the union (Formula presented.) forms a strongly dominating set. A fuzzy coalition partition of (Formula presented.) is defined as (Formula presented.), where each set (Formula presented.) either forms a singleton strongly dominating set or is not a strongly dominating set but forms a fuzzy coalition with another non-strongly dominating set in (Formula presented.). A fuzzy graph with such a fuzzy coalition partition (Formula presented.) is called a fuzzy coalition graph, denoted as (Formula presented.). The vertex set of the fuzzy coalition graph consists of (Formula presented.), corresponding one-to-one with the sets of (Formula presented.), and the two vertices are adjacent in (Formula presented.) if and only if (Formula presented.) and (Formula presented.) are fuzzy coalition partners in (Formula presented.). This study demonstrates how fuzzy coalition graphs can model and optimize cybersecurity collaborations across critical infrastructures in smart cities, ensuring comprehensive protection against cyber threats. This study concludes that fuzzy coalition graphs offer a robust framework for optimizing collaboration and decision-making in interconnected systems like smart cities.