Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inserting an Edge into a Geometric Embedding   The algorithm of Gutwenger et al. to insert an edge $e$ in linear time into a
planar graph $G$ with a minimal number of crossings on $e$, is a helpful tool
for designing heuristics that minimize edge crossings in drawings of general
graphs. Unfortunately, some graphs do not have a geometric embedding $\Gamma$
such that $\Gamma+e$ has the same number of crossings as the embedding $G+e$.
This motivates the study of the computational complexity of the following
problem: Given a combinatorially embedded graph $G$, compute a geometric
embedding $\Gamma$ that has the same combinatorial embedding as $G$ and that
minimizes the crossings of $\Gamma+e$. We give polynomial-time algorithms for
special cases and prove that the general problem is fixed-parameter tractable
in the number of crossings. Moreover, we show how to approximate the number of
crossings by a factor $(\Delta-2)$, where $\Delta$ is the maximum vertex degree
of $G$.

35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin