Ordered Binary Decision Diagrams (OBDDs, in short) are useful representations for Boolean
functions and are applied in various fields such as design and formal verification of digital sys-
tems, combinatorics and artificial intelligence. This paper discusses relationships between the
two OBDDs of a monotone function and of the set of its prime implicants by comparing their
subfunctions. First, we analyze the OBDDs of the characteristic function of stable sets of a
graph, which is a monotone Boolean function, and of the family of all the maximal stable sets,
which corresponds to the set of all the prime implicants of that function. As a result we show
that there exists a monotone function which has an O(n) size DNF but cannot be represented by
a polynomial size OBDD. In other words, we cannot obtain the OBDD of the set of all the prime
implicants of a monotone function in an output-size sensitive manner once we have constructed
the OBDD of that function in the worst case. This gives a negative answer to open questions
by Coudert [3] who has given an implicit method to compute the set of prime implicants of a
Boolean function by using OBDDs for the two-level logic minimization. A positive result is also
given for a meaningful class of matroid functions.