Consider a simple connected fuzzy graph (FG) G and consider an ordered fuzzy subset H = {(u₁, σ(u₁)), (u₂, σ(u₂)), …(u_k, σ(u_k))}, |H| ≥ 2 of a fuzzy graph; then, the representation of σ − H is an ordered k-tuple with regard to H of G. If any two elements of σ − 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∈S, ∃v∈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_γr(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.

Considering a fuzzy graph G is simple and can be connected and considered as a subset $H=\left\{\left(u_1,\sigma \left(u_1\right)\right),\left(u_2,\sigma \left(u_2\right)\right),\dots \left(u_k,\sigma \left(u_k\right)\right)\right\}, \left|H\right|\ge 2$; then, every two pairs of elements of $\sigma -H$ have a unique depiction with the relation of $H$, and $H$ can be termed as a fuzzy resolving set (FRS). The minimal $H$ cardinality is regarded as the fuzzy resolving number (FRN), and it is signified by $Fr\left(G\right)$. An independence set is discussed on the FRS, fuzzy resolving domination set (FRDS), and Fuzzy modified antimagic resolving set (FMARS). In this paper, we discuss the independency of FRS and FMARS in which an application has also been developed.

In this paper, we introduce a new idea in fuzzy graph theory called the Fault-Tolerant Fuzzy Resolving Domination Set (FT-FRDS). This concept combines three important features: fault tolerance (ability to handle errors), resolving power (identifying nodes), and domination (covering or monitoring all nodes). The main contribution of this paper is the introduction of the Fault-Tolerant Fuzzy Resolving Domination Set (FT-FRDS) in fuzzy graphs. It ensures node identification and monitoring even when some nodes fail. The paper defines FT-FRDS, explores its properties, and provides examples to support its practical use. We also provide examples to help understand how it works. This concept is helpful because it makes sure that even if some nodes in the graph fail or stop working, the system can still identify and monitor other nodes correctly. This is especially useful for real-world problems like sensor networks, communication systems, and areas where high reliability is needed despite possible faults or uncertainty.