Miscellaneous Digraph Classes

Abstract

Obviously, there are countless digraph classes, so that any attempt to give a complete overview is doomed to failure. One has to restrict oneself to a selection. This chapter tries to survey some of the digraph classes that were not granted their own chapter in this book. As tournaments are arguably the best studied class of digraphs with a rich library of strong results, it is no wonder that they and their many generalizations are featured prominently throughout the book. In this regard, this chapter poses no exception. We examine arc-locally semicomplete digraphs, which generalize both semicomplete and semicomplete bipartite digraphs, as well as their generalizations \(\mathcal {H}_1\)-free digraphs and \(\mathcal {H}_2\)-free digraphs. The related classes of \(\mathcal {H}_3\)-free digraphs and \(\mathcal {H}_4\)-free digraphs are also briefly considered. Of course, there are also digraph classes (fairly) unrelated to tournaments such as kernel-perfect digraphs which are mentioned in several results throughout the book and appear in this chapter in a section mainly dedicated to perfect digraphs, game-perfect digraphs and weakly game-perfect digraphs. Furthermore, we consider some digraph classes that appear naturally in applications to other fields such as mathematical logic or computer science. Two such classes with applications in the construction of interconnection networks are de Bruijn digraphs and Kautz digraphs. Both classes can be defined using the line digraph operator. We also investigate line digraphs and iterated line digraphs in general. Minimal series-parallel digraphs, series-parallel digraphs and series-parallel partial order digraphs appear in flow diagrams and dependency charts and have an application to the problem of scheduling under constraints. We consider them briefly in a section on directed cographs, a generalization of series-parallel partial order digraphs.