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, and give complete characterizations of the extremal graphs and the extremal minimally 
doi:10.1016/0166-218X(94)90073-6 
Copyright © 1994 Published by Elsevier Science B.V. All rights reserved.
Note
NP-completeness of minimum spanner problems
Received 2 November 1992;
revised 3 May 1993.
Available online 2 April 2002.
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Abstract
A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. This notion is motivated by applications in distributed systems, communication networks, computational geometry and robotics. In this paper, it is shown that for any fixed t ≥ 2, the problem of determining, for a graph G and a positive integer K, whether G contains a t-spanner with at most K edges is NP-complete, even if G is a bipartite graph (for fixed t ≥ 3). The problem for digraphs is also shown to be NP-complete, even for oriented graphs (with fixed t ≥ 3).
