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Let G = (V , E) be a simple connected graph. An ordered subset W of V is said to be a resolving set of G if every vertex is uniquely determined by its vector of distances to the vertices in W. The minimum cardinality of a resolving set is called the resolving number of G and is denoted by r(G). As an extension, the total resolving number was introduced in [6] as the minimum cardinality taken over all resolving sets in which 〈W 〉 has no isolates and it is denoted by tr(G). In this paper, we obtain the bounds on the total resolving number of block graphs and line graphs. Also, we characterize the extremal graphs. AMS subject classification: Primary 05C12, Secondary 05C35.

Total Resolving Number of Block Graphs and Line Graphs