  We introduce the notion of z-topological orderings for digraphs. We prove
that given a digraph G on n vertices admitting a z-topological order- ing,
together with such an ordering, one may count the number of subgraphs of G that
at the same time satisfy a monadic second order formula {\phi} and are the
union of k directed paths, in time f ({\phi}, k, z) * n^O(k*z) . Our result
implies the polynomial time solvability of many natural counting problems on
digraphs admitting z-topological orderings for constant values of z and k.
Concerning the relationship between z-topological orderability and other
digraph width measures, we observe that any digraph of directed path-width d
has a z- topological ordering for z <= 2d + 1. On the other hand, there are
digraphs on n vertices admitting a z-topological order for z = 2, but whose
directed path-width is {\Theta}(log n). Since graphs of bounded directed
path-width can have both arbitrarily large undirected tree-width and
arbitrarily large clique width, our result provides for the first time a
suitable way of partially trans- posing metatheorems developed in the context
of the monadic second order logic of graphs of constant undirected tree-width
and constant clique width to the realm of digraph width measures that are
closed under taking subgraphs and whose constant levels incorporate families of
graphs of arbitrarily large undirected tree-width and arbitrarily large clique
width.
