Fault-tolerant metric basis and dimension of barycentric subdivision of zero divisor graphs

The undirected zero divisor graph of a commutative ring with unity $R$, denoted by $\mathrm{\Gamma }(R)=(V(\mathrm{\Gamma }(R)),E(\mathrm{\Gamma }(R)))$. The vertex set $V(\mathrm{\Gamma }(R))$ consists of all the non-zero zero-divisors of $R$. The edge set $E(\mathrm{\Gamma }(R))$ is defined by the set $\{e=a_1{a}_2\mid a_1\cdot a_2=0\text{ and }{a}_1,a_2\in V(\mathrm{\Gamma }(R))\}$. The barycentric subdivision of $\mathrm{\Gamma }$ is the process of subdividing each edge by inserting new vertex in the graph $\mathrm{\Gamma }$. In this article, we have focused on the fault-tolerant metric dimension of the barycentric subdivision of zero divisor graph of the group of integers modulo $n$, represented by $fdim(BS(\mathrm{\Gamma }({\mathbb{Z}}_n)$, where $n=pq$; $p$ and $q$ are distinct odd primes with $q>p$. We also demonstrate that $fdim(BS(\mathrm{\Gamma }({\mathbb{Z}}_n)\ge q-1$ for every $n=pq$, where $p$ and $q$ are any distinct odd primes with $q>p$.