Inverse fuzzy multigraphs and planarity with application in decision-making

Recently, in Almallah et al. (New Math Nat Comput, to appear) and Borzooei et al. (New Math Nat Comput 16(2):397-418, 2020), we defined the concept of inverse fuzzy graph as a generalization of graph, which is able to answer some problems that graph theory and fuzzy graph theory can not explain. And as we know planarity is very important and applicable concept in graph theory and fuzzy graph theory, so our motivation in this paper is discussion the importance of planarity in inverse fuzzy graphs. Now, in this paper we define the notion of inverse fuzzy multigraph and the concept of planarity on it by using the concepts of intersecting value and inverse fuzzy planarity value. Then we introduce some related theorems which determining upper bounds and lower bounds for the inverse fuzzy planarity value. After that we define the strong (weak) planarity of an inverse fuzzy multigraph and investigate related results. Finally, we give an application of inverse fuzzy multigraphs to decision-making how to reduce the cost of travel tours.