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For an ordered k-decomposition D fG 1 ; G 2 ; . . . ; G k g of a connected graph G and an edge e of G, the D-representation of e is the k-tuple rejD de; G 1 ; de; G 2 ; . . . ; de; G k , where de; G i is the distance from e to G i . A decomposition D is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dec(G). It is shown that for every two positive integers k and n ! 2, there exists a tree T of order n with decT k. It is also shown that decG n for every graph G of order n ! 3 and that decK n 2n 5=3 for n ! 3.

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