  Recently, Kawarabayashi and Thorup presented the first deterministic
edge-connectivity recognition algorithm in near-linear time. A crucial step in
their algorithm uses the existence of vertex subsets of a simple graph $G$ on
$n$ vertices whose contractions leave a multigraph with $\tilde{O}(n/\delta)$
vertices and $\tilde{O}(n)$ edges that preserves all non-trivial min-cuts of
$G$, where $\delta$ is the minimum degree of $G$ and $\tilde{O}$ hides
logarithmic factors.
  We present a simple argument that improves this contraction-based sparsifier
by eliminating the poly-logarithmic factors, that is, we show a
contraction-based sparsification that leaves $O(n/\delta)$ vertices and $O(n)$
edges, preserves all non-trivial min-cuts and can be computed in near-linear
time $\tilde{O}(m)$, where $m$ is the number of edges of $G$. We also obtain
that every simple graph has $O((n/\delta)^2)$ non-trivial min-cuts.
  Our approach allows to represent all non-trivial min-cuts of a graph by a
cactus representation, whose cactus graph has $O(n/\delta)$ vertices. Moreover,
this cactus representation can be derived directly from the standard cactus
representation of all min-cuts in linear time. We apply this compact structure
to show that all min-cuts can be explicitly listed in $\tilde{O}(m) + O(n^2 /
\delta)$ time for every simple graph, which improves the previous best time
bound $O(nm)$ given by Gusfield and Naor.
