  We consider connectivity properties and asymptotic slopes for certain random
directed graphs on $Z^2$ in which the set of points $C_o$ that the origin
connects to is always infinite. We obtain conditions under which the complement
of $C_o$ has no infinite connected component. Applying these results to one of
the most interesting such models leads to an improved lower bound for the
critical occupation probability for oriented site percolation on the triangular
lattice in 2 dimensions.
