Discrepancies between metric dimension and partition dimension of a connected graph

Let (Z2,E4)(Z2,E4) and (Z2,E8)(Z2,E8) be graphs where the set of vertices is the set of points of the integer lattice and the set of edges consists of all pairs of vertices whose city block and chessboard distances, respectively, are 1.In this paper it is shown that the partition dimensions of these graphs are 3 and 4, respectively, while their metric dimensions are not finite. Also, for every n⩾3n⩾3 there exists an induced subgraph of (Z2,E4)(Z2,E4) of order 3n-13n-1 with metric dimension n   and partition dimension 3. These examples will answer a question raised by Chartrand, Salehi and Zhang. Furthermore, graphs of order n⩾9n⩾9 having partition dimension n-2n-2 are characterized, thus completing the characterization of graphs of order n having partition dimension 2, n  , or n-1n-1 given by Chartrand, Salehi and Zhang. The list of these graphs includes 23 members.