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An analysis of different types of arcs and nodes in various types of fuzzy graphs like cycles, trees, blocks, complete fuzzy graphs is made. The notion of convexity in fuzzy graphs is introduced and the existence of a geodesic cover for a connected fuzzy graph is established. The ideas like geodesic iteration number, convex hull, hull number, convexity number, check node, complete node and extreme node of a connected fuzzy graph are introduced. Some thoughts on complement of a fuzzy graph are shared.