Shallow brambles
Nicolas Bousquet, Wouter Cames van Batenburg, Louis Esperet, Gwenaël Joret, Piotr Micek
TL;DR
The paper investigates radius-bounded analogues of classical connectivity parameters in graphs, aiming to relate depth-$r$ bramble number $\mathrm{bn}_r$, depth-$r$ tangle number $\mathrm{tn}_r$, depth-$r$ linkedness $\mathrm{link}_r$, and depth-$r$ well-linkedness $\mathrm{well}_r$ to each other and to sparsity notions like $\omega_r$ and $\mathrm{scol}_r$. It defines depth-$r$ minor-models and brambles, proves a network of polynomially related inequalities linking these parameters and classical ones such as $\omega_r$ and $\mathrm{scol}_r$, and shows these bounds yield polynomial control in monotone classes with polynomial expansion. A key auxiliary result shows that, over all graphs, $\mathrm{scol}_r(G)$ cannot be bounded by any function of $r$ and $\mathrm{bn}_s(G)$, highlighting limits of certain approaches. Overall, the work advances a bounded-radius framework connecting bramble/tangle/linkedness/well-linkedness to classical sparsity invariants and informs whether polynomial expansion implies polynomial bounds on strong coloring numbers.
Abstract
A graph class $\mathcal{C}$ has polynomial expansion if there is a polynomial function $f$ such that for every graph $G\in \mathcal{C}$, each of the depth-$r$ minors of $G$ has average degree at most $f(r)$. In this note, we study bounded-radius variants of some classical graph parameters such as bramble number, linkedness and well-linkedness, and we show that they are pairwise polynomially related. Furthermore, in a monotone graph class with polynomial expansion they are all uniformly bounded by a polynomial in $r$.