A Note on Universal Point Sets for Planar Graphs

Abstract

We investigate which planar point sets allow simultaneous straight-line embeddings of all planar graphs on a fixed number of vertices. We first show that at least $(1.293-o(1))n$ points are required to find a straight-line drawing of each $n$-vertex planar graph (vertices are drawn as the given points); this improves the previous best constant $1.235$ by Kurowski (2004). Our second main result is based on exhaustive computer search: We show that no set of 11 points exists, on which all planar 11-vertex graphs can be simultaneously drawn plane straight-line. This strengthens the result by Cardinal, Hoffmann, and Kusters (2015), that all planar graphs on $n \le 10$ vertices can be simultaneously drawn on particular $n$-universal sets of $n$ points while there are no $n$-universal sets of size $n$ for $n \ge 15$. We also provide 49 planar 11-vertex graphs which cannot be simultaneously drawn on any set of 11 points. This, in fact, is another step towards a (negative) answer of the question, whether every two planar graphs can be drawn simultaneously - a question raised by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw, and Mitchell (2007).