Forbidden Directed Minors, Directed Path-Width and Directed Tree-Width of Tree-Like Digraphs

Abstract

In this paper we consider the directed path-width and directed tree-width of recursively defined digraphs. As an important combinatorial tool, we show how the directed path-width and the directed tree-width can be computed for the disjoint union, order composition, directed union, and series composition of two directed graphs. These results imply the equality of directed path-width and directed tree-width for all digraphs which can be defined by these four operations. This allows us to show a linear-time solution for computing the directed path-width and directed tree-width of all these digraphs. Since directed co-graphs are precisely those digraphs which can be defined by the disjoint union, order composition, and series composition our results imply the equality of directed path-width and directed tree-width for directed co-graphs and also a linear-time solution for computing the directed path-width and directed tree-width of directed co-graphs, which generalizes the known results for undirected co-graphs of Bodlaender and Möhring.We assume that V G ⊆ X e for every e ∈ E T . We define (Whenever this leads to an empty set W ′ s where t is the predecessor of s in T ′ J we remove vertex s from T ′ J and replace every arc (s, t ′ ) by (t, t ′ ) with the corresponding set X (t,t ′ ) = X (s,t ′ ) ∩ V H .is a directed tree-decomposition of H as follows.• W ′ J is a partition of V H into nonempty sets. • Let e be an arc in T ′ J which is also in T J . Since e ∼ s implies W s = W ′ s = {v} for some v ∈ V H normality condition remains true. Arcs (t, t ′ ) in T ′ J which are not in T J are obtained by two arcs (t, s) and (s, t ′ ) from T J . If ∪{W r | r ∈ V T , t ′ ≤ r} is X (s,t ′ ) -normal, then ∪{W r | r ∈ V ′ T , t ′ ≤ r} is X (t,t ′ ) -normal since X (t,t ′ ) = X (s,t ′ ) ∩ V H .

Characterizations for Special Directed Co-graphs

“…The set of all these digraphs is denoted by extended directed co-graphs, since it generalizes directed co-graphs. In [GR18a] we showed that the result of [GR18b] even holds for extended directed co-graphs. For the class of extended directed co-graphs it remains to show how to compute an ex-di-co-tree and to find forbidden subdigraph characterizations.…”

“…In [JRST01] the directed union was introduced as a generalization of the disjoint union and the order composition. In [GR18a] we considered digraphs which can be defined by disjoint union, order composition, directed union, and series composition of two directed graphs. The set of all these digraphs is denoted by extended directed co-graphs, since it generalizes directed co-graphs.…”

Acyclic coloring of special digraphs

“…Furthermore, we conclude that directed co-graphs and the considered generalizations lead to subclasses of perfect digraphs [AH15]. For directed cactus forests, which is a set of digraphs of directed tree-width 1 [GR19b], the results of [Wie19] and [MSW19] lead to an upper bound of 2 for the dichromatic number and that an optimal acyclic coloring can be computed in polynomial time. We show that this even can be done in linear time.…”

Twin-Distance-Hereditary Digraphs

“…We have stated the following Lemma in [15] for directed path-width and directed tree-width. The proof is extendable straight-forward to DAG-width and cycle rank.…”