Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Turing Kernelization for Finding Long Paths in Graph Classes Excluding a
  Topological Minor   The notion of Turing kernelization investigates whether a polynomial-time
algorithm can solve an NP-hard problem, when it is aided by an oracle that can
be queried for the answers to bounded-size subproblems. One of the main open
problems in this direction is whether k-Path admits a polynomial Turing kernel:
can a polynomial-time algorithm determine whether an undirected graph has a
simple path of length k, using an oracle that answers queries of size poly(k)?
  We show this can be done when the input graph avoids a fixed graph H as a
topological minor, thereby significantly generalizing an earlier result for
bounded-degree and $K_{3,t}$-minor-free graphs. Moreover, we show that k-Path
even admits a polynomial Turing kernel when the input graph is not
H-topological-minor-free itself, but contains a known vertex modulator of size
bounded polynomially in the parameter, whose deletion makes it so. To obtain
our results, we build on the graph minors decomposition to show that any
H-topological-minor-free graph that does not contain a k-path, has a separation
that can safely be reduced after communication with the oracle.

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The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: gr-qc
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The categories of this paper are: econ