A study on fuzzy resolving domination sets and their application in network theory

Abstract

Consider a simple connected fuzzy graph (FG) G and consider an ordered fuzzy subset H = {(u(1), sigma(u(1))), (u(2), sigma(u(2))), horizontal ellipsis (u(k), sigma(u(k)))}, |H| >= 2 of a fuzzy graph; then, the representation of sigma - H is an ordered k-tuple with regard to H of G. If any two elements of sigma - H do not have any distinct representation with regard to H, then this subset is called a fuzzy resolving set (FRS) and the smallest cardinality of this set is known as a fuzzy resolving number (FRN) and it is denoted by Fr(G). Similarly, consider a subset S such that for any u is an element of S, there exists v is an element of V - S, then S is called a fuzzy dominating set only if u is a strong arc. Now, again consider a subset F which is both a resolving and dominating set, then it is called a fuzzy resolving domination set (FRDS) and the smallest cardinality of this set is known as the fuzzy resolving domination number (FRDN) and it is denoted by F-gamma r(G). We have defined a few basic properties and theorems based on this FRDN and also developed an application for social network connection. Moreover, a few related statements and illustrations are discussed in order to strengthen the concept.