doubly resolving set of a class of graphs on dihedral group

A set s of vertices of graph γ is a doubly resolving set for γ, if every two distinct vertices of γ are doubly resolved by some two vertices of s. the minimal doubly resolving set of vertices of graph γ is a doubly resolving set with the minimum cardinality and is denoted by ψ(γ). in this paper, we consider the cayley graph cay(d2n, ωk ∪sm), where d2n =< a, b | an = b2 = 1, ba = an−1b > is the dihedral group of order 2n (n ≥ 4), and ω1 = {b, an−1b}, ω2 = ω1 ∪ {ab, an−2b}, ..., ωk = ωk−1 ∪ {ak−1b, an−kb} and s1 = {a, an−1}, s2 = s1 ∪ {a2, an−2}, ..., sm = sm−1 ∪ {am, an−m} are inverse closed subsets of d2n − {1} for any k, m ∈ n, 1 ≤ k, m ≤ [n2 ], and we show that the minimum cardinality of a doubly resolving set of this class of graphs is n in some cases.