  We reduce the problem of finding an augmenting path in a general graph to a
reachability problem in a directed bipartite graph. A slight modification of
depth-first search leads to an algorithm for finding such paths. Although this
setting is equivalent to the traditional terminology of blossoms due to
Edmonds, there are some advantages. Mainly, this point of view enables the
description of algorithms for the solution of matching problems without
explicit analysis of blossoms, nested blossoms, and so on. Exemplary, we
describe an efficient realization of the Hopcroft-Karp approach for the
computation of a maximum cardinality matching in general graphs and a variant
of Edmonds' primal-dual algorithm for the maximum weighted matching problem.
