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Example text

In this case, the two graphs have the same fuzzy vertex set; they differ only in the edge weights. For any fuzzy subset 13 of S such that 13 ~ A, the partial fuzzy subgraph of (A, R) induced by 13 is the maximal partial fuzzy subgraph of (A, R) that has fuzzy vertex set E. Evidently, this is just the partial fuzzy graph (13, T), where T(x, y) = E(x) I\E(y) I\R(x, y) for all x, YES. 1 Paths and Connectedness Let G = (V, X) be a graph. A walk of G is an alternating sequence of vertices and edges vo, xl, VI, ...

Ion of a fuzzy graph, no path from x to x can have strength greater than A( x), which is the strength of the path of length O. Hovvever, it docs not satisfy the triangle property, as we see from the following example. 0 Here any path from x to y or from y to z has strength ~ 1/2 since it must involve either edge (x,y) or edge (y,z). Thus the shortest strongest paths between them have length 1. 1' to z. through u and v, that has length 3 and strength 1. Thus dis(x, z) = 3 > 1 + 1 = dis(x, y) + dis(y, z) in this case.

Then (AI 0 A 2, EI 0 E 2) is a partial fuzzy subgraph of GI [G 2 J. Proof. 25 that (EI 0 E2)((UI, U2)(VI, V2)) ~ (AI 0 A 2)((UI, U2)) /\ (AI 0 A 2)((VI, V2)) for all (UI, U2)(VI, V2) EX. 27 is called the composition of (AI, E I ) with (A 2, E2)' 48 2. 28 Suppose that G is the composition G l [G 2 J of two graphs G l and G2. Let (A, E) be a partial fuzzy subgraph of G. Consider the following equations: (i) Xi I\Yj = A(Vli,V2j),i = 1, ... ,nij = 1, ... ,mi (ii) Xi I\Zjk = E«Vli' V2j)(Vli, V2k)), i = 1, ...