Fault-tolerant metric dimension of the intersection graph of gamma sets in the zero-divisor graph

The stable equivalent set is a finite collection of disjoint vertex subsets for a connected graph [Formula: see text] such that each set induces the same maximal independent set of [Formula: see text] and their union equals [Formula: see text]. The stable equivalence number is the maximum cardinality of the stable equivalent set [Formula: see text], and it is represented by the symbol [Formula: see text]. In order to analyze the fault-tolerant and local metric dimensions of the intersection graph of gamma sets in the zero-divisor graph, [Formula: see text], the stable equivalent set of [Formula: see text] was used.