On the independent metric dimension of some circulant graphs

Let W = {w1, w2, w3, …, wk} be an ordered set of vertices in a connected graph G. The representation of a vertex v ∈ V(G) with respect to W is the k-tuple r(v|W) = (d(v, w1), d(v, w2), …, d(v, wk)), where d(v, wi) is the length of the shortest path from v to wi. If each vertex in G is uniquely identified by the distance vector, r(v|W) = (d(v, w1), d(v, w2),…,d(v, wk)), then W is called a resolving set for G. If the resolving set is also independent, it is referred to as an independent resolving set. The independent metric dimension of G, denoted by idim(G), is the smallest cardinality of an independent resolving set. This study explores the independent metric dimension of the circulant graphs Cn(1, 2), Cn(1, 2, 3), Cn(1, 2, 3, 4) for sufficiently large n.