Menger’s theorem for m-polar fuzzy graphs and application of m-polar fuzzy edges to road network

Abstract

The main objective of this research article is to classify different types of m-polar fuzzy edges in an m-polar fuzzy graph by using the strength of connectedness between pairs of vertices. The identification of types of m-polar fuzzy edges, including α-strong m-polar fuzzy edges, β-strong m-polar fuzzy edges and δ-weak m-polar fuzzy edges proved to be very useful to completely determine the basic structure of m-polar fuzzy graph. We analyze types of m-polar fuzzy edges in strongest m-polar fuzzy path and m-polar fuzzy cycle. Further, we define various terms, including m-polar fuzzy cut-vertex, m-polar fuzzy bridge, strength reducing set of vertices and strength reducing set of edges. We highlight the difference between edge disjoint m-polar fuzzy path and internally disjoint m-polar fuzzy path from one vertex to another vertex in an m-polar fuzzy graph. We define strong size of an m-polar fuzzy graph. We then present the most celebrated result due to Karl Menger for m-polar fuzzy graphs and illustrate the vertex version of Menger’s theorem to find out the strongest m-polar fuzzy paths between affected and non-affected cities of a country due to an earthquake. Moreover, we discuss an application of types of m-polar fuzzy edges to determine traffic-accidental zones in a road network. Finally, a comparative analysis of our research work with existing techniques is presented to prove its applicability and effectiveness.

Connectivity indices of m-polar fuzzy network model, with an application to a product manufacturing problem

“…An m -PF path between two vertices and is said to be an m -PF cycle C (Akram et al. 2021 ) if and . The strength of an m -PF path P is described as .…”

“…(Akram et al. 2021 ) We say that the m -PF graph is a complete bipartite m -PF graph when ( ) such that (i). for each , if or , (ii).…”

“…Other types of edges in m -PF graphs were introduced in Akram et al. ( 2021 ). For other concepts, the readers are suggested to Chen ( 1997 ), Mahapatra and Pal ( 2022 ), Mahapatra et al.…”

“…Obtained the results on connectivity status and status sequence of different structures of FGs Akram et al. ( 2021 ) Menger’s theorem for m -PF graphs 1. Classified different types of m -PF edges in an m PF graph by using the strength of connectedness 2.…”

Directed rough fuzzy graph with application to trade networking

“…Recently, Karunambigai and Buvaneswari ( 2017 ) presented the idea of Menger’s theorem for intuitionistic fuzzy graphs Akram et al. ( 2021 ) discussed Menger’s theorem for m-polar fuzzy graphs and application to road network. In this research study, we present the concept of of vertices and edges in DRFGs which have not yet been explored in the context of DRFGs.…”