Branch-depth is minor closure of contraction-deletion-depth

TL;DR

This work studies matroid branch-depth as a counterpart to graph tree-depth and compares it with contraction-deletion-depth. It proves, for ${\mathbb F}$-representable matroids, that bounded branch-depth is equivalent to being a minor of a matroid with bounded contraction-deletion-depth, via rooted ($d$,$r$)-decompositions and a key Main Lemma that controls subspace structure. The main results yield a precise containment: a class has bounded branch-depth iff it lies in the minor-closure of a class with bounded contraction-deletion-depth, and provide an explicit bound $\mathrm{cdd}(N)\le 2r(4^d-1)+1$ when embedding $M$ into such $N$. The paper also discusses extensions to general matroids and presents open problems and conjectures about tightening the bounds and representability assumptions, highlighting the parallel with tree-depth and shrub-depth in graphs.

Abstract

The notion of branch-depth for matroids was introduced by DeVos, Kwon and Oum as the matroid analogue of the tree-depth of graphs. The contraction-deletion-depth, another tree-depth like parameter of matroids, is the number of recursive steps needed to decompose a matroid by contractions and deletions to single elements. Any matroid with contraction-deletion-depth at most d has branch-depth at most d. However, the two notions are not functionally equivalent as contraction-deletion-depth of matroids with branch-depth two can be arbitrarily large. We show that the two notions are functionally equivalent for representable matroids when minor closures are considered. Namely, an F-representable matroid has small branch-depth if and only if it is a minor of an F-representable matroid with small contraction-deletion-depth. This implies that any class of F-representable matroids has bounded branch-depth if and only if it is a subclass of the minor closure of a class of F-representable matroids with bounded contraction-deletion-depth.

Figures (4)

• Figure 1: The relation of classes of representable matroids with bounded depth and width parameters. The statement of Theorem \ref{['thm:class']} is visualized by the arrow relating classes with bounded contraction-deletion-depth and classes with bounded branch-depth.
• Figure 2: A fat cycle and a graph that contains the depicted fat cycle as a minor. The branch-depth of the graphic matroid associated with either of the graphs is two, the contraction-deletion-depth of the graphic matroid associated with the former graph is six, and the contraction-deletion-depth of the graphic matroid associated with the latter graph is three.
• Figure 3: The ${\mathbb F}$-representation of the matroid $M'$ constructed in the proof of Theorem \ref{['thm:main']} when $k=3$, $d_A=4$, $d'_1=3$, $d_1=4$, $d'_2=2$, $d_2=4$, $d'_3=2$ and $d_3=3$. Stars depict entries that can be arbitrary (both zero or non-zero), and the entries that are not displayed are zero. The representation of the matroid $M"$ from the proof is the part of the representation of $M'$ encompassed by the dashed lines.
• Figure 4: The ${\mathbb F}$-representation of the matroid $N$ constructed in the proof of Theorem \ref{['thm:main']} when $k=3$, $d_A=4$, $d'_1=3$, $d_1=4$, $n_1=1$, $d'_2=2$, $d_2=4$, $n_2=1$, $d'_3=2$, $d_3=3$ and $n_3=2$. Stars depict entries that can be arbitrary (both zero or non-zero), and the entries that are not displayed are zero. The representations of matroids $N_1$, $N_2$ and $N_3$ obtained from induction are encompassed by the dashed lines.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• proof : Proof of Theorem \ref{['thm:bound']}
• proof : Proof of Theorem \ref{['thm:class']}