Balázs Keszegh — Page 5

arXiv2010

Drawing Graphs with Orthogonal Crossings

By a poly-line drawing of a graph G on n vertices we understand a drawing of G in the plane such that each edge is represented by a polygonal arc joining its two respective vertices. We call a turning point of a polygonal arc the bend. We consider the class of graphs that admit a poly-line drawing, in which each edge has at most one bend (resp. two bends) and any two edges can cross only at a right angle. It is shown that the number of edges of such graphs is at most O(n) (resp. O(n \log^2 n)). This is a strengthening of a recent result of Didimo et al.

math.CO↗

arXiv2009

On Polygons Excluding Point Sets

By a polygonization of a finite point set $S$ in the plane we understand a simple polygon having $S$ as the set of its vertices. Let $B$ and $R$ be sets of blue and red points, respectively, in the plane such that $B\cup R$ is in general position, and the convex hull of $B$ contains $k$ interior blue points and $l$ interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points $R$. We show that there is a minimal number $K=K(l)$, which is polynomial in $l$, such that one can always find a blue polygonization excluding all red points, whenever $k\geq K$. Some other related problems are also considered.

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