  We introduce the concept of shallow directed minors and based on this a new
classification of classes of directed graphs which is diametric to existing
directed graph decompositions and width measures proposed in the literature.
  We then study in depth one type of classes of directed graphs which we call
nowhere crownful. The classes are very general as they include, on one hand,
all classes of directed graphs whose underlying undirected class is nowhere
dense, such as planar, bounded-genus, and $H$-minor-free graphs; and on the
other hand, also contain classes of high edge density whose underlying class is
not nowhere dense. Yet we are able to show that problems such as directed
dominating set and many others become fixed-parameter tractable on nowhere
crownful classes of directed graphs. This is of particular interest as these
problems are not tractable on any existing digraph measure for sparse classes.
  The algorithmic results are established via proving a structural equivalence
of nowhere crownful classes and classes of graphs which are directed uniformly
quasi-wide. This rather surprising result is inspired by Nesetril and Ossana de
Mendez (2008) and yet a different and more delicate proof is needed, which is a
significant part of our contribution.
