Cop-width, flip-width and strong colouring numbers
Robert Hickingbotham
TL;DR
This paper proves a tight connection between cop-width and strong colouring numbers by establishing that for every radius $r$, $\operatorname{copwidth}_r(G)\le \,\operatorname{scol}_{4r}(G)$ for all graphs $G$. This bound implies that graph classes with linear strong colouring numbers have linear cop-width and, consequently, linear flip-width, positioning flip-width as a robust dense-graph analogue of generalized colouring numbers. The authors derive improved, explicit bounds for several sparse graph families, notably $K_t$-minor-free graphs and $(g,k)$-planar graphs, and show how these bounds compare favorably to prior weak-colouring-number based results. They also connect bounded expansion with linear cop-width and discuss algorithmic implications via slicewise polynomial approximations for flip-width. Overall, the work advances understanding of how strong colouring numbers constrain dense-graph width parameters and opens avenues for tighter bounds in classical sparse graph classes.
Abstract
Cop-width and flip-width are new families of graph parameters introduced by Toruńczyk (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.