Connected partition dimensions of graphs

For a vertex v of a connected graph G and a subset S of V (G), the distance between v and S is d(v, S) = min{d(v, x)|x ∈ S}. For an ordered k-partition Π = {S1, S2, · · · , Sk} of V (G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v, S1), d(v, S2), · · · , d(v, Sk)). The k-partition Π is a resolving partition if the k-vectors r(v|Π), v ∈ V (G), are distinct. The minimum k for which there is a resolving k-partition of V (G) is the partition dimension pd(G) of G. A resolving partition Π = {S1, S2, · · · , Sk} of V (G) is connected if each subgraph 〈Si〉 induced by Si (1 ≤ i ≤ k) is connected in G. The minimum k for which there is a connected resolving k-partition of V (G) is the connected partition dimension cpd(G) of G. Thus 2 ≤ pd(G) ≤ cpd(G) ≤ n for every connected graph G of order n ≥ 2. The connected partition dimensions of several classes of well-known graphs are determined. It is shown that for every pair a, b of integers with 3 ≤ a ≤ b ≤ 2a − 1, there is a connected graph G having pd(G) = a and cpd(G) = b. Connected graphs of order n ≥ 3 having connected partition dimension 2, n, or n− 1 are characterized.