Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic
  Time   We present faster algorithms for computing the 2-edge and 2-vertex strongly
connected components of a directed graph, which are straightforward
generalizations of strongly connected components. While in undirected graphs
the 2-edge and 2-vertex connected components can be found in linear time, in
directed graphs only rather simple $O(m n)$-time algorithms were known. We use
a hierarchical sparsification technique to obtain algorithms that run in time
$O(n^2)$. For 2-edge strongly connected components our algorithm gives the
first running time improvement in 20 years. Additionally we present an $O(m^2 /
\log{n})$-time algorithm for 2-edge strongly connected components, and thus
improve over the $O(m n)$ running time also when $m = O(n)$. Our approach
extends to k-edge and k-vertex strongly connected components for any constant k
with a running time of $O(n^2 \log^2 n)$ for edges and $O(n^3)$ for vertices.

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The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
19
The categories of this paper are: eess