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 Omega(log(2) 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.