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Total resolving number of edge cycle graphs. (English) Zbl 1468.05062

Summary: 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 \(r(G)\). Total resolving number is the minimum cardinality taken over all resolving sets in which \(\langle W \rangle \) has no isolates and it is denoted by \(tr(G)\). In this paper, we determine the exact values of total resolving number of \(K_{1, n-1} (C_k ), B_{s,t} (C_k ), P_n (C_k ) \) and \(K_n (C_k )\). Also, we obtain bounds for the total resolving number of \(G(C_k )\) when \(G\) is an arbitrary graph and characterize the extremal graphs.

MSC:

05C12 Distance in graphs
05C35 Extremal problems in graph theory
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References:

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