Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bridge-Depth Characterizes which Structural Parameterizations of Vertex
  Cover Admit a Polynomial Kernel   We study the kernelization complexity of structural parameterizations of the
Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing
algorithm that can reduce any instance $(G,k)$ of the Vertex Cover problem to
an equivalent one, whose size is polynomial in the size of a pre-determined
complexity parameter of $G$. A long line of previous research deals with
parameterizations based on the number of vertex deletions needed to reduce $G$
to a member of a simple graph class $\mathcal{F}$, such as forests, graphs of
bounded tree-depth, and graphs of maximum degree two. We set out to find the
most general graph classes $\mathcal{F}$ for which Vertex Cover parameterized
by the vertex-deletion distance of the input graph to $\mathcal{F}$, admits a
polynomial kernelization. We give a complete characterization of the
minor-closed graph families $\mathcal{F}$ for which such a kernelization
exists. We introduce a new graph parameter called bridge-depth, and prove that
a polynomial kernelization exists if and only if $\mathcal{F}$ has bounded
bridge-depth. The proof is based on an interesting connection between
bridge-depth and the size of minimal blocking sets in graphs, which are vertex
sets whose removal decreases the independence number.

12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio