Publication at Faculty of Mathematics and Physics |
2013
Abstract
Motivated by recent papers of Eppstein, Abel et al. and Barat et al., and by a question of Wood, we investigate properties of planar point sets with no 5-hole (no empty convex pentagon). We answer a question of Wood by showing that the visibility graph of a finite point set with no 5-hole may contain a clique of arbitrary size.
This is in contrast with the previous examples of sets with no 5-hole, including the example of the (finite) square lattice. In our construction we use several equivalent local characterizations of (locally) finite planar point sets with no 5-hole which may be of independent interest.
Our construction relies on a construction scheme which allows to derive new, non-trivial examples of sets with no 5-hole.