Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Kuratowski graph planarity criterion   This paper is purely expositional. The statement of the Kuratowski graph
planarity criterion is simple and well-known. However, its classical proof is
not easy. In this paper we present the Makarychev proof (with further
simplifications by Prasolov, Telishev, Zaslavski and the author) which is
possibly the simplest. In the Rusian version before the proof we present all
the necessary definitions, and afterwards we state some close results on graphs
and more general spaces. The paper is accessible for students familiar with the
notion of a graph, and could be an interesting easy reading for mature
mathematicians.

14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc