  This paper is the first from a series of papers that establish a
generalization of the basilica decomposition for cardinality minimum joins in
grafts. Joins in grafts are also known as $T$-joins in graphs, where $T$ is a
given set of vertices, and minimum joins in grafts can be considered as a
generalization of perfect matchings in graphs provided in terms of parity. The
basilica decomposition is a canonical decomposition applicable to general
graphs with perfect matchings, and the general Kotzig-Lov\'asz decomposition is
one of the three central concepts that compose this theory. The classical
Kotzig-Lov\'asz decomposition is a canonical decomposition for a special class
of graphs known as {\em factor-connected graphs} and is famous for its
contribution to the study of the matching polytope and lattice. The general
Kotzig-Lov\'asz decomposition is a nontrivial generalization of its classical
counterpart and is applicable to general graphs with perfect matchings. As a
component of the basilica decomposition theory, the general Kotzig-Lov\'asz
decomposition has contributed to the derivation of further results in matching
theory, such as a characterization of barriers or an alternative proof of the
tight cut lemma. In this paper, we present an analogue of the general
Kotzig-Lov\'asz decomposition for minimum joins in grafts.
