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Minors of plane digraphs

Maria Chudnovsky, Paul Seymour

Abstract

A digraph $H$ is a ``semi-strong minor'' of another, $G$, if a subdivision of $H$ can be obtained from a subdigraph of $G$ by contracting strongly-connected subdigraphs to single vertices. We will define a width measure of ``plane'' digraphs (that is, drawn in the plane) based on a kind of branch-composition, and show that for every plane digraph $H$, all plane digraphs not containing $H$ as a semi-strong minor have bounded width, while plane digraphs in general have unbounded width.

Minors of plane digraphs

Abstract

A digraph is a ``semi-strong minor'' of another, , if a subdivision of can be obtained from a subdigraph of by contracting strongly-connected subdigraphs to single vertices. We will define a width measure of ``plane'' digraphs (that is, drawn in the plane) based on a kind of branch-composition, and show that for every plane digraph , all plane digraphs not containing as a semi-strong minor have bounded width, while plane digraphs in general have unbounded width.

Paper Structure

This paper contains 10 sections, 24 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Drawings
  3. The decomposition
  4. Patterns of directed cuts
  5. Box systems
  6. Paths across a disc
  7. Shortcuts across a cycle
  8. The main result
  9. Grids
  10. Minor containment, and the Kawarabayashi-Kreutzer theorem

Figures (11)

  • Figure 1: A $4\times 4$ grid.
  • Figure 2: The $6\times 6$ diwall.
  • Figure 3: A $(u,v)$-multicut with pattern $(1,1,-1,1,-1)$,
  • Figure 4: The $6\times 6$ semi-grid and alternating grid.
  • Figure 5: The $6\times 6$ diwall, redrawn. If we subdivide once the odd edges of the horizontal paths we obtain a subdigraph of the $k\times 3k$ alternating grid.
  • ...and 6 more figures