The theorems of decomposition and representation for fuzzy graphs (Q1177682)

Using the concepts of exponential graph, \(R(G'')\), and fuzzy exponential graph, \(F(G'')\), a theory on fuzzy graphs is derived. Crisping the fuzzy graph \(\overline{G}\) generates a subgraph of \(G''\). The mechanics of the operations (union, intersection, complementation, etc.) are defined in the classical fuzzy set frame. Two theorems fix properties of the decompositions of the fuzzy graph which permit to express \(\overline{G}\) in terms of strong and weak \(L\)- cut graphs \((G_ L)\) of it by computing \(\bigcup_{L\in[0,1]}LG_ L\). Four definitions involving \(H:[0,1]\to R(G'')\) are used for proving three theorems and two corollaries. They permit the representation of \(F(G'')\) by a class of all classic covers of \(H\) by \(\bigcup_{L\in[0,1]}LH(L)\) and of other related structures and functions. Though the references are at least ten years old the results improve the existing theory in that theme.