Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Towards single face shortest vertex-disjoint paths in undirected planar
  graphs   Given $k$ pairs of terminals $\{(s_{1}, t_{1}), \ldots, (s_{k}, t_{k})\}$ in
a graph $G$, the min-sum $k$ vertex-disjoint paths problem is to find a
collection $\{Q_{1}, Q_{2}, \ldots, Q_{k}\}$ of vertex-disjoint paths with
minimum total length, where $Q_{i}$ is an $s_i$-to-$t_i$ path between $s_i$ and
$t_i$. We consider the problem in planar graphs, where little is known about
computational tractability, even in restricted cases. Kobayashi and Sommer
propose a polynomial-time algorithm for $k \le 3$ in undirected planar graphs
assuming all terminals are adjacent to at most two faces. Colin de Verdiere and
Schrijver give a polynomial-time algorithm when all the sources are on the
boundary of one face and all the sinks are on the boundary of another face and
ask about the existence of a polynomial-time algorithm provided all terminals
are on a common face. We make progress toward Colin de Verdiere and Schrijver's
open question by giving an $O(kn^5)$ time algorithm for undirected planar
graphs when $\{(s_{1}, t_{1}), \ldots, (s_{k}, t_{k})\}$ are in
counter-clockwise order on a common face.

33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph