Abstract
For a constant t≥1, a t-spanner of a connected graph G is a spanning subgraph of G in which the distance between any pair of vertices is at most t times its distance in G. This concept, introduced by Peleg and Ullman in 1989, was used in the construction of an optimal synchronizer for the hypercube. We address the problem of finding a t-spanner with minimum number of edges. This problem is called the minimum t-spanner problem (MinSt), and is known to be ▪-hard for every t≥2 even on bounded-degree graphs. Our main contribution is to improve the previous results, by showing that MinSt is ▪-hard even on planar graphs with maximum degree at most 4 (resp. 5) when t≥4 (resp. t=3). We also show that with a slight modification of a result presented by Kobayashi (2018), MinS2 remains ▪-hard on planar graphs with maximum degree 7.
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