  In the design of greedy algorithms for the maximum cardinality matching
problem the utilization of degree information when selecting the next edge is a
well established and successful approach.
  We define the class of "degree sensitive" greedy matching algorithms, which
allows us to analyze many well-known heuristics, and provide tight
approximation guarantees under worst case tie breaking. We exhibit algorithms
in this class with optimal approximation guarantee for bipartite graphs. In
particular the Karp-Sipser algorithm, which picks an edge incident with a
degree-1 node if possible and otherwise an arbitrary edge, turns out to be
optimal with approximation guarantee D/(2D-2), where D is the maximum degree.
