Fuzzy edge connectivity and fuzzy local edge connectivity with applications to communication networks

Abstract

The authors (Mathew and Sunitha (2010) [9]) introduced the concept of fuzzy arc cuts for improving the definition of the so-called cut-sets proposed by Yeh and Bang, both generalizing edge cuts from classic graph theory to the fuzzy settings. They argued that non-strong edges need not be considered in a fuzzy arc cut as such edges would not change the strength of connectedness between any vertices. However, we give a counterexample in the paper, which illustrates that this viewpoint may contain some flaws. Based on this observation, a more general definition of fuzzy arc cuts (called fuzzy edge cuts) is given. Moreover, we define fuzzy edge connectivity with two parameters, this definition allows us to talk about edge connectivity more precisely. The concepts of fuzzy local edge cuts and fuzzy local edge connectivity are also introduced. We investigate the relations between fuzzy edge cuts and fuzzy local edge cuts. A characterization for the weight of a minimal fuzzy local edge cut is obtained. Finally, we provide some algorithms to compute fuzzy edge connectivity and fuzzy local edge connectivity of special graphs. Besides, applications of two connectivity parameters in communication networks are presented.