On spanners of geometric graphs
Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every t with 1 < t < 1/4 log n, there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n <sup>1+1/t</sup>) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(tn<sup>1+2/(t+1)</sup>) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.
Gudmundsson, J. (Joachim), & Smid, M. (2006). On spanners of geometric graphs. doi:10.1007/11785293_36