Several width measures for digraphs have been proposed in the last few years. However, none of them possess all the “nice” properties of treewidth, namely, (1) being
, that is, admitting polynomial-time algorithms for a large class of problems on digraphs of bounded width; and (2) having nice
such as being monotone under taking subdigraphs and some form of arc contractions. As for (1), MSO
is the least common denominator of all reasonably expressive logical languages that can speak about the edge/arc relation on the vertex set, and so it is quite natural to demand efficient solvability of all MSO
-definable problems in this context. (2) is a necessary condition for a width measure to be characterizable by some version of the cops-and-robber game characterizing treewidth. More specifically, we introduce a notion of a
directed topological minor
and argue that it is the weakest useful notion of minors for digraphs in this context. Our main result states that any
digraph measure that is algorithmically useful and structurally nice cannot be substantially different from the treewidth of the underlying undirected graph.