Forcing geodesic number of a fuzzy graph

Chartrand and Zhang in 1999 introduced the concept of forcing geodetic number of crisp graphs and studied it for several classes of graphs. In this paper, this concept is extended to fuzzy graphs using geodesic distance and is called the forcing geodesic number. For a geodesic basis S of a fuzzy graph G : (V, σ, μ), a subset T of S with the property that S is the unique geodesic basis containing T is called a forcing subset of S. The minimum cardinality of a forcing subset of S is called the forcing geodesic number of S in G and is denoted by gnf (S). The forcing geodesic number of G, denoted by gnf (G), is defined as gnf (G) = min{gnf (S)} where the minimum is taken over all geodesic bases S in G. A characterization of the forcing geodesic number depending on the geodesic bases in the fuzzy graph is identified. The forcing geodesic number of fuzzy trees and of complete fuzzy graphs is obtained. It is proved that if the geodesic number of a fuzzy graph is 2, then its forcing geodesic number is always less than 2.