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Abstract : Let G(V,E) be a connected graph. A subset S of V is said to be 2-resolving set of G, if for every pair of distinct vertices u, v /∈ S, there exists a vertex w ∈ S such that |d(u,w) - d(v,w)| ≥ 2. Among all 2-resolving sets of G, the set having minimum cardinality is called a 2-metric basis of G and its cardinality is called the 2-metric dimension of G and is denoted by βk(G). In this paper, we determine the 2-metric dimension of cartesian product of complete graph with some standard graphs. Further, we have determined the 2-metric dimension of the graphs Pm Pn, Cm Pn and Cm Cn. © 2017 Academic Publications, Ltd.