Subgraphs Satisfying MSO Properties on z-Topologically Orderable Digraphs

We introduce the notion of z-topological orderings for digraphs. We prove that given a digraph G on n vertices admitting a z-topological ordering, 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 φ and are the union of k directed paths, in time f(φ, k, z) · n. 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 ztopological 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 Θ(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 transposing 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.