Optimal Multi-Level Fault-Tolerant Resolving Sets of Circulant Graph C(n : 1, 2)

Let G=(V(G),E(G))
be a simple connected unweighted graph. A set R⊂V(G)
is called a fault-tolerant resolving set with the tolerance level k if the cardinality of the set Sx,y={w∈R:d(w,x)≠d(w,y)}
is at least k for every pair of distinct vertices x,y
of G. A k-level metric dimension refers to the minimum size of a fault-tolerant resolving set with the tolerance level k. In this article, we calculate and determine the k-level metric dimension for the circulant graph C(n:1,2)
for all possible values of k and n.
The optimal fault-tolerant resolving sets with k tolerance are also delineated.