Applied Mathematics & Information Sciences

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A vertex u in a graph G resolves a pair of distinct vertices x, y of G if the distance between u and x is different from the distance between u and y. A set W of vertices in G resolves the graph G if every pair of distinct vertices of G is resolved by some vertices in W. The metric dimension of a graph, denoted by dim(G), is the smallest cardinality of a resolving set. A resolving set W for a connected graph G of order n ¸ 3 is called 2-size resolving set if the size of the subgraph < W > induced by W is two. The minimum cardinality of a 2-size resolving set is called the 2-size metric dimension of G, denoted by tr(G). A 2-size resolving set of cardinality tr(G) is called a tr-set. In this paper, we study 2-size resolving sets in some well-known classes of graphs and give some realizable results.

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