Fuzzy Graph Theory: A Survey

A fuzzy graph (f-graph) is a pair G : (σ, μ) where σ is a fuzzy subset of a set S and μ is a fuzzy relation on σ. A fuzzy graph H : (τ, υ) is called a partial fuzzy subgraph of G : (σ, μ) if τ (u) ≤ σ(u) for every u and υ (u, v) ≤ μ(u, v) for every u and v . In particular we call a partial fuzzy subgraph H : (τ, υ) a fuzzy subgraph of G : (σ, μ ) if τ (u) = σ(u) for every u in τ * and υ (u, v) = μ(u, v) for every arc (u, v) in υ*. A connected f-graph G : (σ, μ) is a fuzzy tree(f-tree) if it has a fuzzy spanning subgraph F : (σ, υ), which is a tree, where for all arcs (x, y) not in F there exists a path from x to y in F whose strength is more than μ(x, y). A path P of length n is a sequence of distinct nodes u0, u1, ..., un such that μ(ui−1, ui) > 0, i = 1, 2, ..., n and the degree of membership of a weakest arc is defined as its strength. If u0 = un and n≥ 3, then P is called a cycle and a cycle P is called a fuzzy cycle(f-cycle) if it contains more than one weakest arc . The strength of connectedness between two nodes x and y is defined as the maximum of the strengths of all paths between x and y and is denoted by CONNG(x, y). An x − y path P is called a strongest x − y path if its strength equals CONNG(x, y). An f-graph G : (σ, μ) is connected if for every x,y in σ ,CONNG(x, y) > 0. In this paper, we offer a survey of selected recent results on fuzzy graphs.