Directed Nowhere Dense Classes of Graphs

Abstract

Many natural computational problems on graphs such as finding dominating or independent sets of a certain size are well known to be intractable, both in the classical sense as well as in the framework of parameterized complexity. Much work therefore has focussed on exhibiting restricted classes of graphs on which these problems become tractable. While in the case of undirected graphs, there is a rich structure theory which can be used to develop tractable algorithms for these problems on large classes of undirected graphs, such a theory is much less developed for directed graphs. Many attempts to identify structure properties of directed graphs tailored towards algorithmic applications have focussed on a directed analogue of undirected tree-width. These attempts have proved to be successful in the development of algorithms for linkage problems but none of the existing width-measures allow for tractable solutions to important problems such as dominating sets and many other related problems. In this paper we take a radically different approach to identifying classes of directed graphs where domination and other problems become tractable. In particular, whereas most existing approaches treat the class of acyclic graphs as simple in their respective width measure, we will specifically study classes of digraphs which do not contain all acyclic digraphs. It is this new approach that make the algorithmic results reported herein possible. More specifically, 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 directed 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 the 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. While this result is inspired by [Nešetřil and Ossona de Mendez 2008], their proof method does not extend to the directed case and a different and much more involved proof is needed, turning it into a particularly significant part of our contribution.