On the metric dimension of corona product graphs

Given a set of vertices S={v"1,v"2,...,v"k} of a connected graph G, the metric representation of a vertex v of G with respect to S is the vector r(v|S)=(d(v,v"1),d(v,v"2),...,d(v,v"k)), where d(v,v"i), i@?{1,...,k} denotes the distance between v and v"i. S is a resolving set for G if for every pair of distinct vertices u,v of G, r(u|S) r(v|S). The metric dimension of G, dim(G), is the minimum cardinality of any resolving set for G. Let G and H be two graphs of order n"1 and n"2, respectively. The corona product G@?H is defined as the graph obtained from G and H by taking one copy of G and n"1 copies of H and joining by an edge each vertex from the ith-copy of H with the ith-vertex of G. For any integer k>=2, we define the graph G@?^kH recursively from G@?H as G@?^kH=(G@?^k^-^1H)@?H. We give several results on the metric dimension of G@?^kH. For instance, we show that given two connected graphs G and H of order n"1>=2 and n"2>=2, respectively, if the diameter of H is at most two, then dim(G@?^kH)=n"1(n"2+1)^k^-^1dim(H). Moreover, if n"2>=7 and the diameter of H is greater than five or H is a cycle graph, then dim(G@?^kH)=n"1(n"2+1)^k^-^1dim(K"1@?H).