Improving Upper and Lower Bounds for the Total Number of Edge Crossings of Euclidean Minimum Weight Laman Graphs

We investigate the total number of edge crossings (i.e., the crossing number) of the Euclidean minimum weight Laman graph MLG(P) on a planar point set P. Bereg et al. (2016) showed that the upper and lower bounds for the crossing number of MLG(P) are 6 vertical bar P vertical bar - 9 and vertical bar P vertical bar - 3, respectively. In this paper, we improve these upper and lower bounds given by Bereg et al. (2016) to 2.5 vertical bar P vertical bar - 5 and (1.25 - epsilon)vertical bar P vertical bar for any epsilon > 0, respectively. Especially, for improving the upper bound, we introduce a novel counting scheme based on some geometric observations.