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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 thatAmong 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 β k (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.

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Abstract. For a simple, connected, undirected graph neighbor-hood coloring of the graph

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