Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Generalised Colouring Numbers on Classes of Bounded Expansion   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.

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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: q-bio
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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph