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TopThe notion of fuzzy graph was introduced in 1975 by Kochen (Kochen, 1975). Fuzzy analogues of many structures in crisp graph theory, like bridges, cut nodes, connectedness, trees and cycles were developed after that.
Fuzzy trees were characterized by Sunitha and Vijayakumar (Sunitha & Vijayakumar, 1999). The authors have characterized fuzzy trees using its unique maximum spanning tree. A sufficient condition for a node to be a fuzzy cut node is also established. Center problems in fuzzy graphs, blocks in fuzzy graphs and properties of self-complementary fuzzy graphs were also studied by the same authors. They have obtained a characterization for blocks in fuzzy graphs using the concept of strongest paths (Sunitha & Vijayakumar, 2005).
Bhutani and Rosenfeld (Bhutani & Rosenfeld, 2003) have introduced the concepts of strong arcs, fuzzy end nodes and geodesics in fuzzy graphs. The authors have defined the concepts of strong arcs and strong paths.
As far as the applications are concerned (information networks, electric circuits, etc.), the reduction of flow between pairs of nodes is more relevant and may frequently occur than the total disruption of the flow or the disconnection of the entire network.