Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Graphical Representation and Hierarchical Decomposition Mechanism for
  Vertex-Cover Solution Space   In this paper, solution space organization of minimum vertex-cover problem is
deeply investigated using the K\"{o}nig-Eg\'{e}rvary (KE) graph and theorem, in
which a hierarchical decomposition mechanism named KE-layer structure of
general graphs is proposed to reveal the complexity of vertex-cover. An
algorithm to verify the KE graph is given by the solution space expression of
vertex-cover, and the relation between multi-layer KE graphs and maximal
matching is illustrated and proved. Furthermore, a framework to calculate the
KE-layer number and approximate the minimal vertex-cover is provided, with
different strategies of switching nodes and counting energy. The phase
transition phenomenon between different KE-layers are studied with the
transition points located, and searching of vertex-cover got by this strategy
presents comparable advantage against several other methods. The graphical
representation and hierarchical decomposition provide a new perspective to
illustrate the intrinsic complexity for large-scale graphs/systems recognition.

16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex