In [12] it was shown that nowhere dense classes of graphs admit sparse neighbourhood covers of small degree. We show that a monotone graph class admits sparse neighbourhood covers if and only if it is nowhere dense. The existence of such covers for nowhere dense classes is established through bounds on so-called weak colouring numbers. The core results of this paper are various lower and upper bounds on the weak colouring numbers and other, closely related generalised colouring numbers. We prove tight bounds for these numbers on graphs of bounded treewidth. We clarify and tighten the relation between the density of shallow minors and the various generalised colouring numbers. These upper bounds are complemented by new, stronger exponential lower bounds on the weak and strong colouring numbers, and by super-polynomial lower bounds on the weak colouring numbers on classes of polynomial expansion. Finally, we show that computing weak r-colouring numbers is NP-complete for all r ≥ 3. Introduction.Nowhere dense classes of graphs have been introduced by Nešetřil and Ossona de Mendez [18,20] as a general model of uniformly sparse graph classes. They include and generalise many other natural sparse graph classes, among them all classes of bounded degree, classes of bounded genus, classes defined by excluded (topological) minors, and classes of bounded expansion. It has been demonstrated in several papers, e.g., [2,8,12,16,18] that nowhere dense graph classes have nice algorithmic properties; many problems that are hard in general can be solved (more) efficiently on nowhere dense graph classes. In fact, nowhere dense classes are a natural limit for the efficient solvability of a wide class of problems [7,12,15].In [12], it was shown that nowhere dense classes of graphs admit sparse neighbourhood covers. Neighbourhood covers play an important role in the study of distributed network algorithms and other application areas (see for example [21]). The neighbourhood covers developed in [12] combine low radius and low degree making them interesting for the applications outlined above. In this paper, we prove a (partial) converse to the result of [12]: we show that monotone graph classes (that is, classes closed under taking subgraphs) are nowhere dense if and only if they admit sparse neighbourhood covers. A similar characterisation result was proved for classes of bounded expansion [19].Nowhere denseness has turned out to be a very robust property of graph classes with various seemingly unrelated characterisations (see [11,18]), among them characterisations through so-called generalised colouring numbers. These are particularly relevant in the algorithmic context, because the existence of sparse neighbourhood covers for nowhere dense

This paper examines the supply response of the Greek pork market. A GARCH process is used to estimate expected price and price volatility, while price and supply equations are estimated jointly. In addition to the standard GARCH model, several different symmetric, asymmetric, and nonlinear GARCH models are estimated. The empirical results indicate that among the estimated GARCH models, the quadratic NAGARCH model seems to better describe producers' price volatility, which was found to be an important risk factor of the supply response function of the Greek pork market. Furthermore, the empirical findings show that feed price is an important cost factor of the supply response function and that high uncertainty restricts the expansion of the Greek pork sector. Finally, the model provides forecasts for quantity supplied, producers' price, and price volatility.

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We show that a graph class G has bounded expansion if and only if it has bounded r-neighbourhood complexity, i.e. for any vertex set X of any subgraph H of G ∈ G, the number of subsets of X which are exact r-neighbourhoods of vertices of H on X is linear in the size of X. This is established by bounding the r-neighbourhood complexity of a graph in terms of both its r-centred colouring number and its weak r-colouring number, which provide known characterisations to the property of bounded expansion.

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