G.Ghorai, M.Pal: Applications of bipolar fuzzy sets in interval graphs

Abstract

Currently, bipolar fuzzy graph is a growing research topic as it is the generalization of fuzzy graphs. In this paper, normal and convex bipolar fuzzy sets are defined and the notion of bipolar fuzzy interval is introduced as a generalization of fuzzy interval and described various methods of their construction. It is shown that intersection of two bipolar fuzzy intervals may not be a bipolar fuzzy interval. Finally, bipolar fuzzy interval graphs is introduced as the intersection graph of a finite family of bipolar fuzzy intervals. The relationship between the intersection graph of a {α, β}-level family of bipolar fuzzy intervals and {α, β}-cut of the intersection graph for that family have been established. It is proved that for every bipolar fuzzy interval graph G, the (α, β)-cut level graph Gᵅᵦ is an interval graph for each (α, β) ∈ (0, 1] × [−1, 0). Also, some important hereditary properties of bipolar fuzzy interval graphs are presented.