Efficient algorithms and edge crossing properties of Euclidean minimum weight Laman graphs

We investigate the Euclidean minimum weight Laman graph on a planar point set P, MLG(P) for short. Bereg et al. (2016) studied geometric proper-ties of MLG(P) and showed that the upper and lower bounds for the total number of edge crossings in MLG(P) are 6|P| - 9 and |P| - 3, respectively. In this paper, we improve these upper and lower bounds to 2.5|P| - 5 and (1.25 - e)|P| for any e > 0, respectively. For improving the upper bound, we introduce a novel counting scheme based on some geometric observations. We also propose an O(|P|(2)) time algorithm for computing MLG(P), which was regarded as one of interesting future works by Bereg et al. (2016).