Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cut Tree Structures with Applications on Contraction-Based
  Sparsification   We introduce three new cut tree structures of graphs $G$ in which the vertex
set of the tree is a partition of $V(G)$ and contractions of tree vertices
satisfy sparsification requirements that preserve various types of cuts.
Recently, Kawarabayashi and Thorup \cite{Kawarabayashi2015a} presented the
first deterministic near-linear edge-connectivity recognition algorithm. A
crucial step in this algorithm uses the existence of vertex subsets of a simple
graph $G$ whose contractions leave a graph with $\tilde{O}(n/\delta)$ vertices
and $\tilde{O}(n)$ edges ($n := |V(G)|$) such that all non-trivial min-cuts of
$G$ are preserved. We improve this result by eliminating the poly-logarithmic
factors, that is, we show a contraction-based sparsification that leaves
$O(n/\delta)$ vertices and $O(n)$ edges and preserves all non-trivial min-cuts.
We complement this result by giving a sparsification that leaves $O(n/\delta)$
vertices and $O(n)$ edges such that all (possibly not minimum) cuts of size
less than $\delta$ are preserved, by using contractions in a second tree
structure. As consequence, we have that every simple graph has $O(n/\delta)$
$\delta$-edge-connected components, and, if it is connected, it has
$O((n/\delta)^2)$ non-trivial min-cuts. All these results are proven to be
asymptotically optimal.
  By using a third tree structure, we give a new lower bound on the number of
\emph{pendant pairs}. The previous best bound was given 1974 by Mader, who
showed that every simple graph contains $\Omega(\delta^2)$ pendant pairs. We
improve this result by showing that every simple graph $G$ with $\delta \geq 5$
or $\lambda \geq 4$ or $\kappa \geq 3$ contains $\Omega(\delta n)$ pendant
pairs. We prove that this bound is asymptotically tight from several
perspectives, and that $\Omega(\delta n)$ pendant pairs can be computed
efficiently.

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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph