Fuzzy Graphs

Abstract

A graph is a pair (V, R), where V is a set and R is a relation on V. The elements of V are thought of as vertices of the graph and the elements of R are thought of as the edges Similarly, any fuzzy relation ρ on a fuzzy subset μ of a set V can be regarded as defining a weighted graph, or fuzzy graph, where the edge (x, y) ∈ V × V has weight or strength ρ(x, y) ∈ [0, 1]. In this chapter, we shall use graph terminology and introduce fuzzy analogs of several basic graph-theoretical concepts. For simplicity, we will consider only undirected graphs through out this chapter unless otherwise specified. Therefore, all fuzzy relations are symmetric and all edges are regarded as unordered pairs of vertices. We abuse notation by writing (x, y) for an edge in an undirected graph (V, R), where x, y ∈ V. (We need not consider loops, that is, edges of the form (x, x); we can assume, if we wish, that our fuzzy relation is reflexive.) Formally, a fuzzy graph G = (V, μ, ρ) is a nonempty set V together with a pair of functions μ: V → [0, 1] and μ: V × V → [0, 1] such that for all x, y in V, ρ(x, y) ≤ μ(x) ∧ μ(y). We call μ the fuzzy vertex set of G and ρ the fuzzy edge set of G, respectively. Note that ρ is a fuzzy relation on μ. We will assume that, unless otherwise specified, the underlying set is V and that it is finite. Therefore, for the sake of notational convenience, we omit V in the sequel and use the notation G = (μ,ρ). Thus in the most general case, both vertices and edges have membership value. However, in the special case where μ(x) = 1, for all x ∈ V, edges alone have fuzzy membership. So, in this case, we use the abbreviated notation G = (V, ρ). The fuzzy graph H = (v,τ) is called a partial fuzzy subgraph of G = (μ, ρ) if v ⊆ μ and τ ⊆ ρ. Similarly, the fuzzy graph H = (P, v, τ) is called a fuzzy subgraph of G = (V, μ, ρ) induced by P if P ⊆ V, v(x) = μ(x) for all x ∈ P and τ(x, y) = ρ(x, y) for all x, y ∈ P. For the sake of simplicity, we sometimes call H a fuzzy subgraph of G. It is worth noticing that a fuzzy subgraph (P, v, τ) of a fuzzy graph (V, μ, ρ) is in fact a special case of a partial fuzzy subgraph obtained as follows.