Degree constrained tree embedding into points in the plane
Information Processing Letters
One strike against the min-max degree triangulation problem
Computational Geometry: Theory and Applications
On embedding an outer-planar graph in a point set
Computational Geometry: Theory and Applications
Tight degree bounds for pseudo-triangulations of points
Computational Geometry: Theory and Applications - Special issue: The European workshop on computational geometry -- CG01
Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons
SIAM Journal on Discrete Mathematics
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Let S be a set of n points in general position in the plane. Together with S we are given a set of parity constraints, that is, every point of S is labeled either even or odd. A graph G on S satisfies the parity constraint of a point p *** S , if the parity of the degree of p in G matches its label. In this paper we study how well various classes of planar graphs can satisfy arbitrary parity constraints. Specifically, we show that we can always find a plane tree, a two-connected outerplanar graph, or a pointed pseudo-triangulation which satisfy all but at most three parity constraints. With triangulations we can satisfy about 2/3 of all parity constraints. In contrast, for a given simple polygon H with polygonal holes on S , we show that it is NP-complete to decide whether there exists a triangulation of H that satisfies all parity constraints.