Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 2-Level Quasi-Planarity or How Caterpillars Climb (SPQR-)Trees   Given a bipartite graph $G=(V_b,V_r,E)$, the $2$-Level Quasi-Planarity
problem asks for the existence of a drawing of $G$ in the plane such that the
vertices in $V_b$ and in $V_r$ lie along two parallel lines $\ell_b$ and
$\ell_r$, respectively, each edge in $E$ is drawn in the unbounded strip of the
plane delimited by $\ell_b$ and $\ell_r$, and no three edges in $E$ pairwise
cross.
  We prove that the $2$-Level Quasi-Planarity problem is NP-complete. This
answers an open question of Dujmovi\'c, P\'{o}r, and Wood. Furthermore, we show
that the problem becomes linear-time solvable if the ordering of the vertices
in $V_b$ along $\ell_b$ is prescribed. Our contributions provide the first
results on the computational complexity of recognizing quasi-planar graphs,
which is a long-standing open question.
  Our linear-time algorithm exploits several ingredients, including a
combinatorial characterization of the positive instances of the problem in
terms of the existence of a planar embedding with a caterpillar-like structure,
and an SPQR-tree-based algorithm for testing the existence of such a planar
embedding. Our algorithm builds upon a classification of the types of
embeddings with respect to the structure of the portion of the caterpillar they
contain and performs a computation of the realizable embedding types based on a
succinct description of their features by means of constant-size gadgets.

36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga