On point-sets that support planar graphs

On Point-Sets That Support Planar Graphs

Article history: Received 14 October 2011 Accepted 27 March 2012 Available online xxxx Communicated by D. Wagner

On Universal Point Sets for Planar Graphs

A set P of points in R 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 col...

Straight Line Embeddings of Planar Graphs on Point Sets

Small Point Sets for Simply-Nested Planar Graphs

A point set P ⊆ R is universal for a class G if every graph of G has a planar straight-line embedding into P . We prove that there exists a O(n( logn log log n )) size universal point set for the class of simply-nested n-vertex planar graphs. This is a step towards a full answer for the well-known open problem on the size of the smallest universal point sets for planar graphs [1,5,9].

Point sets with planar embeddings of cubic, connected graphs

Let P be a set of n ≥ 3 points in the plane in general position with n even and let h being the number of points on the convex hull of P . We want to know in which cases P admits a planar embedding of a cubic, connected graph (a “cubic embedding”)? García et al. proved in [1] that cubic embeddings always exists for h ≤ 4n, but for higher values of h the question remained open. We present a char...