On Spanners of Geometric Graphs [chapter]

Joachim Gudmundsson, Michiel Smid
2006 Lecture Notes in Computer Science  
Given a connected geometric graph G, we consider the problem of constructing a tspanner of G having the minimum number of edges. We prove that for every real number 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 1+1/t ) 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(n 1+2/(t−1) ) edges. We also prove that the
more » ... lem 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.
doi:10.1007/11785293_36 fatcat:dq7uksrjgjbrfjye7vjd6yj6me