The Generalised Colouring Numbers on Classes of Bounded Expansion

TL;DR

This paper explores the properties of generalized colouring numbers in bounded expansion graph classes, establishing their relation with other measures and applying these insights to prove fixed-parameter tractability of certain logical model-checking problems.

Contribution

It introduces the concept of universal orders for graphs excluding fixed topological minors, linking colouring numbers with other graph measures, and provides a new proof for fixed-parameter tractability results.

Findings

Graphs excluding fixed topological minors admit universal orders.

Universal orders ensure small colouring numbers for all r.

Model-checking for successor-invariant first-order formulas is fixed-parameter tractable.

Abstract

The generalised colouring numbers $\mathrm{adm}_r(G)$, $\mathrm{col}_r(G)$, and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as generalisations of the usual colouring number, also known as the degeneracy of a graph, and have since then found important applications in the theory of bounded expansion and nowhere dense classes of graphs, introduced by Ne\v{s}et\v{r}il and Ossona de Mendez. In this paper, we study the relation of the colouring numbers with two other measures that characterise nowhere dense classes of graphs, namely with uniform quasi-wideness, studied first by Dawar et al. in the context of preservation theorems for first-order logic, and with the splitter game, introduced by Grohe et al. We show that every graph excluding a fixed topological minor admits a universal order, that is, one order witnessing that the colouring numbers are small for every value of $r$. Finally, we use our construction of such orders to give a new proof of a result of Eickmeyer and Kawarabayashi, showing that the model-checking problem for successor-invariant first-order formulas is fixed-parameter tractable on classes of graphs with excluded topological minors.