Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How many vertex locations can be arbitrarily chosen when drawing planar
  graphs?   It is proven that every set $S$ of distinct points in the plane with
cardinality $\lceil \frac{\sqrt{\log_2 n}-1}{4} \rceil$ can be a subset of the
vertices of a crossing-free straight-line drawing of any planar graph with $n$
vertices. It is also proven that if $S$ is restricted to be a one-sided convex
point set, its cardinality increases to $\lceil \sqrt[3]{n} \rceil$. The proofs
are constructive and give rise to O(n)-time drawing algorithms. As a part of
our proofs, we show that every maximal planar graph contains a large induced
biconnected outerplanar graphs and a large induced outerpath (an outerplanar
graph whose weak dual is a path).

2
The categories of this paper are: math
19
The categories of this paper are: eess
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin
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The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg