Universal point sets for planar graphs

Importance: High ✭✭✭
Author(s): Mohar, Bojan
Recomm. for undergrads: no
Posted by: mdevos
on: May 22nd, 2007

We say that a set $P \subseteq {\mathbb R}^2$ is $n$-\emph{universal} if every $n$ vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in $P$, and all edges are (non-intersecting) straight line segments.

\begin{question} Does there exist an $n$-universal set of size $O(n)$? \end{question}

More generally, if we let $f(n)$ denote the size of the smallest $n$-universal set, we are interested in the behaviour of $f$. The best known upper bound is $f(n) = O(n^2)$. Indeed, every $n$-vertex planar graph can be drawn as required in the $n \times n$ grid [dFPP], [S]. On the flip side, it is known that $f(n) \ge 1.098n$ for sufficiently large $n$ [CH].

Bibliography

[CH] M. Chrobak and H.Karloff. A lower bound on the size of universal sets for planar graphs. SIGACT News, 20:83-86, 1989.

[dFPP] H. de Fraysseix, J. Pach, and R. Pollack. \href[How to draw a planar graph on a grid]{http://www.springerlink.com/content/b471830661k10534/}. Combinatorica, 10(1):41-51, 1990. \MRhref{1075065}

[S] W. Schnyder. Embedding planar graphs on the grid. In Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pages 138-148, 1990.


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