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 3 4 ⌊ n 4 ⌋ ≤ ψ ( n ) ≤ ⌈ n 4 ⌉ . In this paper we prove that ψ ( 13 ) = 3 , which improves the upper bound on ψ ( n ) to ⌈ 3 n 13 ⌉ , thus answering a question posed by Aichholzer et al. in the same paper.