NP-Completeness Results for Minimum Planar Spanners

For any fixed parameter t greater or equal to 1, a \emph t-spanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A \emph minimum t-spanner is a t-spanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this paper, we prove the NP-hardness of finding minimum t-spanners for planar weighted graphs and digraphs if t greater or equal to 3, and for planar unweighted graphs and digraphs if t greater or equal to 5. We thus extend results on that problem to the interesting case where the instances are known to be planar. We also introduce the related problem of finding minimum \emphplanar t-spanners and establish its NP-hardness for similar fixed values of t.