The partition 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 vertical bar S) = (d(v, v(1)), d(v, v(2)), d(v, v(k))), where d(v, v(i)), i is an element of {1,..., k} denotes the distance between v and v(i). S is a resolving set of G if for every pair of distinct vertices u, v of G, r(u vertical bar S) not equal r(v vertical bar S). The metric dimension dim(G) of G is the minimum cardinality of any resolving set of G. Given an ordered partition Pi = {P-1, P-2,..., P-t} of vertices of a connected graph G, the partition representation of a vertex v of G, with respect to the partition Pi is the vector r(v/Pi) = (d(v, P-1), d(v, P-2),..., d(v, P-t)), where d(v, P-i), 1 <= i <= t, represents the distance between the vertex v and the set Pi, that is d(v, Pi) = min(u is an element of Pi){d(v, u)}. Pi is a resolving partition for G if for every pair of distinct vertices u, v of G, r(u vertical bar Pi) not equal r(v vertical bar Pi). The partition dimension pd(G) of G is the minimum number of sets in any resolving partition for G. Let G and H be two graphs of order n(1) and n(2) respectively. The corona product G circle dot H is defined as the graph obtained from G and H by taking one copy of G and n(1) copies of H and then joining by an edge, all the vertices from the ith-copy of H with the ith-vertex of G. Here we study the relationship between pd(G OH) and several parameters of the graphs G circle dot H, G and H, including dim(G circle dot H), pd(G) and pd(H).