The transitive reduction of a binary relation 
on a set 
is the minimum relation 
on 
with the same transitive
closure as 
.
Thus 
for any elements 
and 
of 
,
provided that 
and there exists no element 
of 
such that 
and 
.
The transitive reduction of a graph 
is the smallest graph 
such that 
, where 
is the transitive closure
of 
(Skiena 1990, p. 203).
