To appear in Semigroup forum

For an arbitrary set X (?nite or in?nite), denote by T (X) the semigroup of full transforma-
tions on X. For ? ? T (X), let C(?) = {? ? T (X) : ?? = ??} be the centralizer of ? in
T (X). The aim of this paper is to characterize the elements of C(?). The characterization is
obtained by decomposing ? as a join of connected partial transformations on X and analyzing
the homomorphisms of the directed graphs representing the connected transformations. The
paper closes with a number of open problems and suggestions of future investigations.