  We provide an analogue of the Dulmage-Mendelsohn decomposition for a class of
grafts known as comb-bipartite grafts. The Dulmage-Mendelsohn decomposition in
matching theory is a classical canonical structure theorem for bipartite
graphs. The substantial part of this classical theorem resides in bipartite
graphs that are factorizable, that is, those with a perfect matching. Minimum
joins in grafts, also known as minimum $T$-joins in graphs,
  is a generalization of perfect matchings in factorizable graphs. Seb\"o
revealed in his paper that comb-bipartite grafts form one of the two
fundamental classes of grafts that serve as skeletons or building blocks of any
grafts. Particularly, any bipartite grafts, that is, bipartite counterpart of
grafts, can be considered as a recursive combination of comb-bipartite grafts.
In this paper, we generalize the Dulmage-Mendelsohn decomposition for
comb-bipartite grafts. We also show for this decomposition a property that is
characteristics to grafts using the general Kotzig-Lov\'asz decomposition for
grafts, which is a known graft analogue of another canonical structure theorem
from matching theory. This paper is the first from a series of studies
regarding bipartite grafts.
