  Given a directed graph D = (N, A) and a sequence of positive integers 1 <=
c_1 < c_2 < ... < c_m <= |N|, we consider those path and cycle polytopes that
are defined as the convex hulls of simple paths and cycles of D of cardinality
c_p for some p, respectively. We present integer characterizations of these
polytopes by facet defining linear inequalities for which the separation
problem can be solved in polynomial time. These inequalities can simply be
transformed into inequalities that characterize the integer points of the
undirected counterparts of cardinality constrained path and cycle polytopes.
Beyond we investigate some further inequalities, in particular inequalities
that are specific to odd/even paths and cycles.
