On the metric dimension of corona product graphs

Given a set of vertices S={v1,v2,…,vk}S={v1,v2,…,vk} of a connected graph GG, the metric representation of a vertex vv of GG with respect to SS is the vector r(v|S)=(d(v,v1),d(v,v2),…,d(v,vk))r(v|S)=(d(v,v1),d(v,v2),…,d(v,vk)), where d(v,vi)d(v,vi), i∈{1,…,k}i∈{1,…,k} denotes the distance between vv and vivi. SS is a resolving set for GG if for every pair of distinct vertices u,vu,v of GG, r(u|S)≠r(v|S)r(u|S)≠r(v|S). The metric dimension of GG, dim(G)dim(G), is the minimum cardinality of any resolving set for GG. Let GG and HH be two graphs of order n1n1 and n2n2, respectively. The corona product G⊙HG⊙H is defined as the graph obtained from GG and HH by taking one copy of GG and n1n1 copies of HH and joining by an edge each vertex from the iith-copy of HH with the iith-vertex of GG. For any integer k≥2k≥2, we define the graph G⊙kHG⊙kH recursively from G⊙HG⊙H as G⊙kH=(G⊙k−1H)⊙HG⊙kH=(G⊙k−1H)⊙H. We give several results on the metric dimension of G⊙kHG⊙kH. For instance, we show that given two connected graphs GG and HH of order n1≥2n1≥2 and n2≥2n2≥2, respectively, if the diameter of HH is at most two, then dim(G⊙kH)=n1(n2+1)k−1dim(H)dim(G⊙kH)=n1(n2+1)k−1dim(H). Moreover, if n2≥7n2≥7 and the diameter of HH is greater than five or HH is a cycle graph, then dim(G⊙kH)=n1(n2+1)k−1dim(K1⊙H)dim(G⊙kH)=n1(n2+1)k−1dim(K1⊙H).