Directed branch-width: A directed analogue of tree-width

TL;DR

This work develops directed branch-width ($\mathrm{dbw}$) as a true directed analogue of tree-width and proves a tight equivalence: a digraph class $\mathcal{C}$ has bounded $\mathrm{dbw}$ if and only if the class of directed line graphs $\vec{L}(\mathcal{C})$ has bounded bi-cut-rank-width ($\mathrm{bcrk}$). The core technical contribution is the dual bounding: $\mathrm{bcrk}(\vec{L}(D)) \le 2(\mathrm{dbw}(D)+1)$ and $\mathrm{dbw}(D) \le 8(1+2^{\mathrm{bcrk}(\vec{L}(D))})$, established via a consistency-labeling framework. The authors relate $\mathrm{dbw}$ to the underlying undirected branch-width, show that $\mathrm{dbw}$ can be small even when $\mathrm{bw}(u(D))$ is large, and prove that $\mathrm{dbw}$ is closed under butterfly minors but not under directed topological minors. Algorithmically, several problems become fixed-parameter tractable when parameterized by $\mathrm{dbw}$, including $\mathrm{D}$-Hamilton-Path, $\mathrm{D}$-Max-Cut, and restricted MSO$_2$ model checking, though they also demonstrate limits (e.g., $\mathrm{MSO}_2$ model checking without the source-sink invariance can be hard). Overall, this width notion provides a principled middle ground between undirected tree-width and directed connectivity, enabling tractable algorithms for a broader class of digraphs and guiding future work on directed tangles and meta-theorems.

Abstract

Gurski and Wanke showed that a graph class C has bounded tree-width if and only if its associated class of directed line graphs has bounded clique-width. Inevitably -- asking whether this relationship lifts to directed graphs -- we introduce a new digraph width measure: we obtain it by investigating digraphs whose directed line graphs have bounded cliquewidth. Thus, to generalize Gurski and Wanke's aforementioned result, we introduce a natural generalization of branch-width to digraphs and we name it accordingly. Directed branch-width is a genuinely directed width-measure insofar as it cannot be used to bound the value of the underlying undirected tree-width. Despite this, the two measures are still closely related: the directed branch-width of a digraph D can differ from the branch-width of its underlying undirected graph only at sources and sinks. This relationship allows us to extend a range of algorithmic results from directed graphs with bounded underlying treewidth to the strictly larger class of digraphs having bounded directed branch-width.

Figures (2)

• Figure 1: An orientation $D$ of a $(3 \times 3)$-grid (left) and a directed branch decompositions of this grid (right). Letting $X = \{ \overrightarrow{eh},\overrightarrow{ih},\overrightarrow{if},\overrightarrow{fe} \}$, the edge $\xi$ is associated with the edge partitions $(E(G) \setminus X, X)$ and $(X, E(G) \setminus X)$. These partitions are themselves respectively associated with the directed vertex separators $\{ e \}$ and $\{ e, f \}$.
• Figure 2: Left: the graph $D_{3}$ defined in the proof of Theorem \ref{['thm:dbw_not_topo_minor_closed']} (the relevant $2$-contractible edges are drawn red and dotted). Right: the graph $\Delta'_{3}$.

Theorems & Definitions (77)

• Theorem 1
• definition 1
• definition 2
• definition 3
• definition 4: robertsonX
• Theorem 2: robertsonX
• definition 5
• definition 6: oum2006approximating
• definition 7: kante2011
• Theorem 3: gurski2016directedNLC, oum2006approximating