Fault-tolerant metric dimension of zero-divisor graphs of commutative rings

Abstract

Let R be a commutative ring with identity. The zero-divisor graph of R denoted by (Formula presented.) is an undirected graph (Formula presented.) where (Formula presented.) is the set of non-zero zero-divisors of R and there is an edge between the vertices z 1 and z 2 in (Formula presented.) if (Formula presented.) A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. If we remove any vertex in a resolving set, then the resulting set is also a resolving set, called the fault-tolerant resolving set, and its minimum cardinality is called the fault-tolerant metric dimension. In this article, we study the fault-tolerant metric dimension for (Formula presented.) where R = (Formula presented.) and (Formula presented.) Furthermore, we obtain some results regarding the line graph of (Formula presented.) and the zero-divisor graph of a Cartesian product of fields.