  A connected graph, on four or more vertices, is matching covered if every
edge is present in some perfect matching. An ear decomposition theorem (similar
to the one for $2$-connected graphs) exists for bipartite matching covered
graphs due to Hetyei. From the results and proofs of Lov\'asz and Plummer, that
rely on Hetyei's theorem, one may deduce that any minimal bipartite matching
covered graph has at least $2(m-n+2)$ vertices of degree two (where minimal
means that deleting any edge results in a graph that is not matching covered);
such a graph is said to be extremal if it attains the stated lower bound.
  In this paper, we provide a complete characterization of the class of
extremal minimal bipartite matching covered graphs. In particular, we prove
that every such graph $G$ is obtained from two copies of a tree devoid of
degree two vertices, say $T$ and $T'$, by adding edges -- each of which joins a
leaf of $T$ with the corresponding leaf of $T'$.
  Apart from the aforementioned bound, there are four other bounds that appear
in, or may be deduced from, the work of Lov\'asz and Plummer. Each of these
bounds leads to a notion of extremality. In this paper, we obtain a complete
characterization of all of these extremal classes and also establish
relationships between them. Two of our characterizations are in the same spirit
as the one stated above. For the remaining two extremal classes, we reduce each
of them to one of the already characterized extremal classes using standard
matching theoretic operations.
