  We consider the minimum cut problem in undirected, weighted graphs. We give a
simple algorithm to find a minimum cut that $2$-respects (cuts two edges of) a
spanning tree $T$ of a graph $G$. This procedure can be used in place of the
complicated subroutine given in Karger's near-linear time minimum cut algorithm
(J. ACM, 2000). We give a self-contained version of Karger's algorithm with the
new procedure, which is easy to state and relatively simple to implement. It
produces a minimum cut on an $m$-edge, $n$-vertex graph in $O(m \log^3 n)$ time
with high probability, matching the complexity of Karger's approach.
