Shallow brambles
Nicolas Bousquet, Wouter Cames van Batenburg, Louis Esperet, Gwena\"el Joret, Piotr Micek

TL;DR
This paper investigates bounded-radius variants of classical graph parameters within classes of graphs with polynomial expansion, establishing their polynomial relationships and bounds.
Contribution
It introduces bounded-radius variants of bramble number, linkedness, and well-linkedness, proving their polynomial relationships and bounds in polynomial expansion graph classes.
Findings
Bounded-radius parameters are pairwise polynomially related.
In polynomial expansion classes, these parameters are uniformly bounded by a polynomial in radius.
The study extends classical graph parameters to localized variants with polynomial bounds.
Abstract
A graph class has polynomial expansion if there is a polynomial function such that for every graph , each of the depth- minors of has average degree at most . 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 .
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph theory and applications · Advanced Combinatorial Mathematics
