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Directed treewidth is closed under taking butterfly minors

Gunwoo Kim, Meike Hatzel, Stephan Kreutzer

Abstract

Butterfly minors are a generalisation of the minor containment relation for undirected graphs to directed graphs. Many results in directed structural graph theory use this notion as a central tool next to directed treewidth, a generalisation of the width measure treewidth to directed graphs. Adler [JCTB'07] showed that the directed treewidth is not closed under taking butterfly minors. Over the years, many alternative definitions for directed treewidth appeared throughout the literature, equivalent to the original definition up to small functions. In this paper, we consider the major ones and show that not all of them share the problem identified by Adler.

Directed treewidth is closed under taking butterfly minors

Abstract

Butterfly minors are a generalisation of the minor containment relation for undirected graphs to directed graphs. Many results in directed structural graph theory use this notion as a central tool next to directed treewidth, a generalisation of the width measure treewidth to directed graphs. Adler [JCTB'07] showed that the directed treewidth is not closed under taking butterfly minors. Over the years, many alternative definitions for directed treewidth appeared throughout the literature, equivalent to the original definition up to small functions. In this paper, we consider the major ones and show that not all of them share the problem identified by Adler.

Paper Structure

This paper contains 19 sections, 45 theorems, 2 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Butterfly models
  4. Cops-and-robber games
  5. Obstructions
  6. $k$-linked sets
  7. Havens
  8. Brambles
  9. Brambles are closed under butterfly minors
  10. Definitions of directed tree decompositions
  11. Overview of results
  12. Directed treewidth that is closed under taking butterfly minors
  13. SC_0 is not a strict upper bound on NCW
  14. Strategy trees
  15. The SC_0 can be strictly less than the NCW-directed treewidth
  16. ...and 4 more sections

Key Result

Lemma 2.2

Let $D$ be a digraph and $k \geq 1$.

Figures (8)

  • Figure 1: The digraph $D$ on the left is a butterfly minor of the digraph $D'$ to the right. However, $D$ contains a weak bramble of order $2$ while $D'$ has no weak bramble of order $2$.
  • Figure 2: The relation between directed tree-width with respect to different types of directed tree decompositions. An arrow with '$\leq$' means bounded in one direction, and a bidirected arrow with '$\lesseqgtr$' means not bounded in any direction.
  • Figure 3: In the proof of \ref{['lem:Adler_lemma1']}, the left figure in $\mathcal{T}$ is replaced by the right one in $\mathcal{T'}$, where a new node $t'$ is added after $t$. Additionally, one of the vertices in the bag of $t$ is split off to form the new bag of $t'$.
  • Figure 4: The digraph $D_2$ from the proof of \ref{['thm:NCW<SCE']} with $\mathsf{NCW}(D_2) < \mathsf{SC}_{\emptyset}(D_2)$. The digraph is originally given by Adler adler_directed_2007.
  • Figure 5: An $\mathsf{NCW}$-directed tree decomposition of $D_2$ in \ref{['fig:D2']} of width $3$, implying $\mathsf{NCW}(D_2) \leq 3$.
  • ...and 3 more figures

Theorems & Definitions (103)

  • Lemma 2.2: johnson_directed_2001
  • Lemma 2.3: Reed reed_introducing_1999
  • Lemma 2.4: Johnson, Robertson, Seymour, Thomas johnson_directed_2001
  • Theorem 2.5: johnson_directed_2001
  • Lemma 2.6: reed_introducing_1999
  • Lemma 2.7: safari_d-width_2005
  • Lemma 2.8: safari_d-width_2005
  • Corollary 2.9
  • Lemma 3.2
  • proof
  • ...and 93 more