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Θ k-graphs are geometric graphs that appear in the context of graph navigation. The shortest-path metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TD-Delaunay graphs, a.k.a. triangular-distance Delaunay triangulations introduced by Chew, have been shown(More)
The family of well-orderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a well-orderly map. We show that the number of well-orderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled(More)
We consider the question: " What is the smallest degree that can be achieved for a plane spanner of a Euclidean graph E ? " The best known bound on the degree is 14. We show that E always contains a plane spanner of maximum degree 6 and stretch factor 6. This spanner can be constructed efficiently in linear time given the Triangular Distance Delaunay(More)
A geometric graph is angle-monotone if every pair of ver-tices has a path between them that—after some rotation—is x-and y-monotone. Angle-monotone graphs are √ 2-spanners and they are increasing chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle monotone graphs. We give a(More)
We use Schnyder woods of 3-connected planar graphs to produce convex straight line drawings on a grid of size (n − 2 − ∆) × (n − 2 − ∆). The parameter ∆ ≥ 0 depends on the Schnyder wood used for the drawing. This parameter is in the range 0 ≤ ∆ ≤ n 2 − 2. The algorithm is a refinement of the face-counting-algorithm, thus, in particular, the size of the grid(More)