In this paper, the edge version of the geodesic number of a fuzzy graph is introduced and the properties satisfied are identified. A comparison between the vertex and edge version of the geodesic number of fuzzy graphs is obtained. The edge geodesic number of fuzzy trees, complete fuzzy graphs, complete bipartite fuzzy graphs and of fuzzy cycles are identified. A necessary and sufficient condition for the existence of an edge geodesic cover in a fuzzy graph is obtained. An application of edge geodesic sets in transportation systems in optimizing the number of traffic inspectors patrolling an urban road network is demonstrated. The fuzziness in the problem helps to identify routes receiving less priority among passengers, elimination of which minimizes the loss suffered by various transport corporations due to lack of collection.
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