On pseudo-convex partitions of a planar point set

Aichholzer et al. [O. Aichholzer, C. Huemer, S. Kappes, B. Speckmann, C.D. Tóth, Decompositions, partitions, and coverings with convex polygons and pseudo-triangles, Graphs and Combinatorics 23 (2007) 481-507] introduced the notion of pseudo-convex partitioning of planar point sets and proved that the pseudo-convex partition number ψ(n) satisfies 34⌊n4⌋≤ψ(n)≤⌈n4⌉. In this paper we prove that ψ(13)=3, which improves the upper bound on ψ(n) to ⌈3n13⌉, thus answering a question posed by Aichholzer et al. in the same paper.