Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How much does a treedepth modulator help to obtain polynomial kernels
  beyond sparse graphs?   In the last years, kernelization with structural parameters has been an
active area of research within the field of parameterized complexity. As a
relevant example, Gajarsk{\`y} et al. [ESA 2013] proved that every graph
problem satisfying a property called finite integer index admits a linear
kernel on graphs of bounded expansion and an almost linear kernel on nowhere
dense graphs, parameterized by the size of a $c$-treedepth modulator, which is
a vertex set whose removal results in a graph of treedepth at most $c$, where
$c \geq 1$ is a fixed integer. The authors left as further research to
investigate this parameter on general graphs, and in particular to find
problems that, while admitting polynomial kernels on sparse graphs, behave
differently on general graphs.
  In this article we answer this question by finding two very natural such
problems: we prove that Vertex Cover admits a polynomial kernel on general
graphs for any integer $c \geq 1$, and that Dominating Set does not for any
integer $c \geq 2$ even on degenerate graphs, unless $\text{NP} \subseteq
\text{coNP}/\text{poly}$. For the positive result, we build on the techniques
of Jansen and Bodlaender [STACS 2011], and for the negative result we use a
polynomial parameter transformation for $c\geq 3$ and an OR-cross-composition
for $c = 2$. As existing results imply that Dominating Set admits a polynomial
kernel on degenerate graphs for $c = 1$, our result provides a dichotomy about
the existence of polynomial kernels for Dominating Set on degenerate graphs
with this parameter.

7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
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The categories of this paper are: eess
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
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The categories of this paper are: q-fin