Improved ▪-hardness results for the minimum

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 ( MinS t ), 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 MinS t 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), MinS 2 remains ▪-hard on planar graphs with maximum degree 7. • Minimum t-spanner, for t ≥ 4, on planar graphs G with Δ(G) ≤ 4 is NP-hard. • Minimum 3-spanner on planar graphs G with Δ(G) ≤ 5 is NP-hard. • Minimum 2-spanner on planar graphs G with Δ(G) ≤ 7 is NP-hard.