  The Dulmage--Mendelsohn decomposition (or the DM-decomposition) gives a
unique partition of the vertex set of a bipartite graph reflecting the
structure of all the maximum matchings therein. A bipartite graph is said to be
DM-irreducible if its DM-decomposition consists of a single component.
  In this paper, we focus on the problem of making a given bipartite graph
DM-irreducible by adding edges. When the input bipartite graph is balanced
(i.e., both sides have the same number of vertices) and has a perfect matching,
this problem is equivalent to making a directed graph strongly connected by
adding edges, for which the minimum number of additional edges was
characterized by Eswaran and Tarjan (1976).
  We give a general solution to this problem, which is divided into three
parts. We first show that our problem can be formulated as a special case of a
general framework of covering supermodular functions, which was introduced by
Frank and Jord\'an (1995) to investigate the directed connectivity augmentation
problem. Secondly, when the input graph is not balanced, the problem is solved
via matroid intersection. This result can be extended to the minimum cost
version in which the addition of an edge gives rise to an individual cost.
Thirdly, for balanced input graphs, we devise a combinatorial algorithm that
finds a minimum number of additional edges to attain the DM-irreducibility,
while the minimum cost version of this problem is NP-hard. These results also
lead to min-max characterizations of the minimum number, which generalize the
result of Eswaran and Tarjan.
