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

In a fuzzy graph G, a fuzzy coalition is formed by two disjoint vertex sets V1 and V2, neither of which is a strongly dominating set, but the union V1∪V2 forms a strongly dominating set. A fuzzy coalition partition of G is defined as Π={V1,V2,⋯,Vk}, where each set Vi 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 Π. A fuzzy graph with such a fuzzy coalition partition Π is called a fuzzy coalition graph, denoted as FG(G,Π). The vertex set of the fuzzy coalition graph consists of {V1,V2,⋯,Vk}, corresponding one-to-one with the sets of Π, and the two vertices are adjacent in FG(G,Π) if and only if Vi and Vj are fuzzy coalition partners in Π. 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.