Geodesic
On Universal Point Sets for Planar Graphs
Journal of Graph Algorithms and Applications, Tome 19 (2015) no. 1, pp. 529-547.

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A set P of points in R2 is n-universal if every planar graph on n vertices admits a plane straight-line embedding on P. Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n ≥ 15. Conversely, we use a computer program to show that there exist universal point sets for all n ≤ 10 and to enumerate all corresponding order types. Finally, we describe a collection G of 7′393 planar graphs on 35 vertices that do not admit a simultaneous geometric embedding without mapping, that is, no set of 35 points in the plane supports a plane straight-line embedding of all graphs in G.
DOI : 10.7155/jgaa.00374
Keywords: universal point set, simultaneous geometric embedding, order type 
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Jean Cardinal; Michael Hoffmann; Vincent Kusters. On Universal Point Sets for Planar Graphs. Journal of Graph Algorithms and Applications, Tome 19 (2015) no. 1, pp. 529-547. doi : 10.7155/jgaa.00374. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00374/

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