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(MATH) Let $\overlinecr(<i>G</i>)$ denote the rectilinear crossing number of a graph <i>$G</i>. We determine $\overlinecr(<i>K</i> <inf>11</inf>)=102 and $\overlinecr(<i>K</i> <inf>12</inf>)=153. Despite the remarkable hunt for crossing numbers of the complete graph <i>.K</i> <inf>n</inf> -- initiated by R. Guy in the 1960s -- these quantities have been(More)
The farthest line segment Voronoi diagram shows properties different from both the closest-segment Voronoi diagram and the farthest-point Voronoi diagram. Surprisingly, this structure did not receive attention in the computational geometry literature. We analyze its combinatorial and topological properties and outline an O(n log n) time construction(More)
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We extend the order type data base of all realizable order types in the plane to point sets of cardinality 11. More precisely, we provide a complete data base of all combinatorial different sets of up to 11 points in general position in the plane. In addition, we develop a novel and efficient method for a complete extension to order types of size 12 and(More)
Let T S be the set of all crossing-free straight line spanning trees of a planar n-point set S. Consider the graph T S where two members T and T of T S are adjacent if T intersects T only in points of S or in common edges. We prove that the diameter of T S is O(log k), where k denotes the number of convex layers of S. Based on this result, we show that the(More)
We investigate the number of plane geometric, i.e., straight-line, graphs, a set <i>S</i> of <i>n</i> points in the plane admits. We show that the number of plane graphs and connected plane graphs as well as the number of cycle-free plane graphs is minimized when <i>S</i> is in convex position. Moreover, these results hold for all these graphs with an(More)