[Paper Review] Computing directed path-width and directed tree-width of recursively defined digraphs

Frank Gurski, Carolin Rehs|arXiv (Cornell University)|Jun 12, 2018

Advanced Graph Theory Research29 references4 citations

TL;DR

This paper presents a linear-time algorithm for computing directed path-width and directed tree-width in recursively defined digraphs, including directed co-graphs, by establishing their equality through constructive decomposition rules for disjoint union, order composition, directed union, and series composition. The key contribution is a generalization of known results for undirected co-graphs to the directed setting, enabling efficient computation for a broad class of 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 Moehring.

Motivation & Objective

• To extend the known equality of path-width and tree-width in undirected co-graphs to the directed setting.
• To develop a constructive method for computing directed path-width and directed tree-width in recursively defined digraphs.
• To generalize results from undirected co-graphs to directed co-graphs and extended directed co-graphs.
• To provide a linear-time algorithm for computing these width parameters in directed co-graphs.
• To establish theoretical foundations for width parameters in directed graphs using decomposition operations.

Proposed method

• The paper defines four operations—disjoint union, order composition, directed union, and series composition—on directed graphs to recursively build complex digraphs.
• It proves that for each of these operations, the directed path-width and directed tree-width of the resulting digraph can be computed from the parameters of the component digraphs.
• Constructive proofs are provided to build directed path-decompositions and directed tree-decompositions from the decompositions of the components.
• The paper introduces the bidirectional complete subdigraph and bipartite subdigraph lemmas to support the decomposition rules.
• It establishes that directed path-width and directed tree-width are equal for all digraphs defined by the four operations.
• The method is extended to directed co-graphs and extended directed co-graphs by showing that they are closed under the four operations.

Experimental results

Research questions

• RQ1 Can directed path-width and directed tree-width be computed efficiently for recursively defined digraphs using decomposition rules?
• RQ2 Do directed path-width and directed tree-width remain equal for digraphs constructed via disjoint union, order composition, directed union, and series composition?
• RQ3 Is the equality of directed path-width and directed tree-width preserved in directed co-graphs, generalizing the undirected co-graph result?
• RQ4 Can a linear-time algorithm be constructed for computing both width parameters in directed co-graphs?
• RQ5 How do the width parameters of a digraph relate to those of its strong components and acyclic condensation?

Key findings

• The directed path-width and directed tree-width are equal for all digraphs constructible via disjoint union, order composition, directed union, and series composition.
• A linear-time algorithm exists for computing both directed path-width and directed tree-width in such digraphs, including directed co-graphs.
• For directed co-graphs, the directed path-width and directed tree-width are equal to the size of the largest bioriented clique minus one.
• The directed path-width and directed tree-width of a general digraph are bounded by the maximum of these parameters over its strongly connected components.
• The results generalize prior work on undirected co-graphs by Bodlaender and Möhri, extending the equality of path-width and tree-width to the directed case.
• The approach is robust under different definitions of directed tree-width, including those allowing empty sets in the decomposition, except for Reed’s definition which may yield higher values.