On k-convex point sets

O. Aichholzer, F. Aurenhammer, T. Hackl, F. Hurtado, A. Pilz*, P. Ramos, J. Urrutia, P. Valtr, B. Vogtenhuber

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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(log2 n) 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 ≥ 3
Original languageEnglish
Pages (from-to)809-832
JournalComputational Geometry: Theory and Applications
Issue number8
Publication statusPublished - 2014

Fields of Expertise

  • Information, Communication & Computing

Treatment code (Nähere Zuordnung)

  • Theoretical


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