On path partition dimension of trees

For a connected graph G and any two vertices u and v in G, let d(u, v) denote the distance between u and v and let d(G) be the diameter of G. For a subset S of V(G), the distance between v and S is d(v, S) = min{d(v, x) vertical bar x epsilon S}. Let Pi = {S-1, S-2, ..., S-k} be an ordered k-partition of V(G). The representation of v with respect to II is the k-vector r(v vertical bar Pi) = (d(v, S-1), d(v, S-2), ... d(v, S-k)). A partition II is a resolving partition for G if the k-vectors r(v vertical bar Pi), v epsilon V(G) are distinct. The minimum k for which there is a resolving k-partition of V(G) is the partition dimension of G, and is denoted by pd(G). A partition Pi = {S-1, S-2, ..., S-k} is a resolving path k-partition for G if it is a resolving partition and each subgraph induced by S-i, 1 <= i <= k, is a path. The minimum k for which there exists a path resolving k-partition of V(G) is the path partition dimension of G, denoted by ppd(G). In this paper path partition dimensions of trees and the existence of graphs with given path partition, partition and metric dimension, respectively are studied.