Article ID Journal Published Year Pages File Type
415252 Computational Geometry 2014 24 Pages PDF
Abstract

We extend the (recently introduced) notion of k-convexity of a two-dimensional subset of the Euclidean plane to finite point sets. A set of n points is considered k-convex if there exists a spanning (simple) polygonization such that the intersection of any straight line with its interior consists of at most k disjoint intervals. As the main combinatorial result, we show that every n  -point set contains a subset of Ω(log2n) points that are in 2-convex position. This bound is asymptotically tight. From an algorithmic point of view, we show that 2-convexity of a finite point set can be decided in polynomial time, whereas the corresponding problem on k  -convexity becomes NP-complete for any fixed k≥3k≥3.

Related Topics
Physical Sciences and Engineering Computer Science Computational Theory and Mathematics
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