Abstract
A path P connecting a pair of vertices in a connected fuzzy graph is called a fuzzy detour, if its μ - length is maximum among all the feasible paths between them. In this paper we establish the notion of fuzzy detour convex sets, fuzzy detour covering, fuzzy detour basis, fuzzy detour number, fuzzy detour blocks and investigate some of their properties. It has been proved that, for a complete fuzzy graph G, the set of any pair of vertices in G is a fuzzy detour covering. A necessary and sufficient condition for a complete fuzzy graph to become a fuzzy detour block is also established. It has been proved that for a fuzzy tree there exists a nested chain of sets, where each set is a fuzzy detour convex. Application of fuzzy detour covering and fuzzy detour basis is also presented.
Highlights
IntroductionRosenfeld’s papers encouraged researchers to think differently and gave them freedom to explore
A complete fuzzy graph, fuzzy detour covering and fuzzy detour basis in association with applications of blockchain [13] based solution can considered to be appropriate for reducing intermediaries
In this paper, we presented the notions of fuzzy detour convex set, detour covering, detour basis and detour number for a fuzzy graph and illustrated them with suitable examples
Summary
Rosenfeld’s papers encouraged researchers to think differently and gave them freedom to explore In fuzzy graphs, he developed fuzzy relations on fuzzy sets and verified graph theoretic concepts. Johns and Songlin Tian in [10] It was Rosenfeld who presented a metric called the μ – distance [11] in fuzzy graphs, later, which was effectively used by various authors in their studies. Umamaheswari presented the notion of fuzzy detour μ – distance and some of its properties in [1]. Proposition 2.7 For the fuzzy detour μ – distance ∆ on a connected fuzzy graph G: (V, σ, μ), (V(G), ∆ ). Proposition 2.8 Every fuzzy detour in a complete fuzzy graph is a Hamiltonian path. Proposition 2.9 A complete fuzzy graph G with n vertices has at least n(n−1) distinct fuzzy detour paths
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