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Obstructions and dualities for matroid depth parameters

Jakub Gajarský, Kristýna Pekárková, Michał Pilipczuk

Abstract

Contraction$^*$-depth is considered to be one of the analogues of graph tree-depth in the matroid setting. In this paper, we investigate structural properties of contraction$^*$-depth of matroids representable over finite fields and rationals. In particular, we prove that the obstructions for contraction$^*$-depth for these classes of matroids are bounded in size. From this we derive analogous results for related notions of contraction-depth and deletion-depth. Moreover, we define a dual notion to contraction$^*$-depth, named deletion$^*$-depth, for $\mathbb{F}$-representable matroids, and by duality extend our results from contraction$^*$-depth to this notion.

Obstructions and dualities for matroid depth parameters

Abstract

Contraction-depth is considered to be one of the analogues of graph tree-depth in the matroid setting. In this paper, we investigate structural properties of contraction-depth of matroids representable over finite fields and rationals. In particular, we prove that the obstructions for contraction-depth for these classes of matroids are bounded in size. From this we derive analogous results for related notions of contraction-depth and deletion-depth. Moreover, we define a dual notion to contraction-depth, named deletion-depth, for -representable matroids, and by duality extend our results from contraction-depth to this notion.
Paper Structure (26 sections, 36 theorems, 35 equations, 1 figure)

This paper contains 26 sections, 36 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work.
  3. Preliminaries
  4. Graph theory
  5. Linear algebra and matrices
  6. Matroids
  7. Matroid contraction and deletion
  8. Matroid duality
  9. Matroid parameters and their properties
  10. Logic
  11. Expressing contraction$^*$-depth in CMSO
  12. Rephrasing the definition of contraction$^*$-depth
  13. Virtual columns
  14. Linear dependence in quotient spaces in CMSO
  15. Formula for contraction$^*$-depth
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\mathbb{F}$ be a finite field. Let $M$ be an $\mathbb{F}$-represented matroid such that $\mathop{\mathrm{c^{*}\space d}}\nolimits (M) = d$ and $\mathop{\mathrm{c^{*}\space d}}\nolimits (M \setminus e) < d$ for every $e \in M$. Then $|M| \leqslant f(|\mathbb{F}|, d)$ for some function $f$.

Figures (1)

  • Figure 1: An example of a matrix $A$ and its associated structure $\mathcal{S}(A)$.

Theorems & Definitions (62)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Lemma 8: Deletion is dual to contraction
  • Lemma 9
  • Definition 10: DeVKO20
  • ...and 52 more