Digraph decompositions and monotonicity in digraph searching

Graph decompositions such as tree-decompositions and associated width measures have been the focus of much attention in structural and algorithmic graph theory. In particular, it has been found that many otherwise intractable problems become tractable on graph classes of bounded tree-width. More recently, proposals have been made to define a similar notion to tree-width for directed graphs. Several proposals have appeared so far, supported by algorithmic applications. In this paper we explore the limits of algorithmic applicability of digraph decompositions and show that various natural candidates for problems, which potentially could benefit from digraphs having small “directed width”, remain NP-complete even on almost acyclic graphs. Closely related to graph and digraph decompositions are graph searching games. An important property of graph searching games is monotonicity and a large number of papers addresses the question whether particular variants of these games are monotone. However, so far for two natural types of graph searching games–underlying DAG- and Kelly-decompositions–the question whether they are monotone was still open. We settle this issue by showing that both variants, the visible and the inert invisible graph searching games on directed graphs, are non-monotone.