(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)
The number of minimum pseudo-triangulations is minimized for point sets in convex position.
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)