Cop-width, flip-width and strong colouring numbers

TL;DR

This paper introduces bounds on cop-width and flip-width graph parameters using strong colouring numbers, showing that classes with linear strong colouring numbers also have linear cop-width and flip-width, with implications for sparse graph classes.

Contribution

It establishes the first bounds relating cop-width and flip-width to strong colouring numbers, generalizing several known graph parameters.

Findings

Bounded cop-width and flip-width in terms of strong colouring numbers.

Linear strong colouring numbers imply linear cop-width and flip-width.

Improved bounds for sparse graph classes.

Abstract

Cop-width and flip-width are new families of graph parameters introduced by Toru\'nczyk (2023) that generalise treewidth, degeneracy, generalised colouring numbers, clique-width and twin-width. In this paper, we bound the cop-width and flip-width of a graph by its strong colouring numbers. In particular, we show that for every $r\in \mathbb{N}$, every graph $G$ has $\text{copwidth}_r(G)\leq \text{scol}_{4r}(G)$. This implies that every class of graphs with linear strong colouring numbers has linear cop-width and linear flip-width. We use this result to deduce improved bounds for cop-width and flip-width for various sparse graph classes.