It was shown by Grohe et al. 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 tree width. We clarify and tighten the relation between the expansion (in the sense of "bounded expansion" as defined by Nesetril and Ossona de Mendez) and the various generalised colouring numbers. These upper bounds are complemented by new, stronger exponential lower bounds on the generalised colouring numbers. Finally, we show that computing weak r-colouring numbers is NP-complete for all r>2.
The Generalised Colouring Numbers on Classes of Bounded Expansion
广义着色数$\mathrm{adm}_r(G)$、$\mathrm{col}_r(G)$和$\mathrm{wcol}_r(G)$是由Kierstead和Yang作为通常着色数的推广而引入的，也被称为图的退化度，并且从那时起在Nešetzil和Ossona de Mendez引入的有界扩张和无处稠密类图的理论中找到了重要的应用。本文研究了色数与另外两种测度的关系，即与均匀拟宽的测度的关系，这两种测度是Dawar等人在一阶逻辑的保持定理中首先研究的，以及与Grohe等人引入的分裂器博弈的关系。我们证明了除固定拓扑子图外的每个图都有一个泛序，即，一阶证明对于$r$的每个值，着色数都很小。最后，我们使用我们的建设这样的订单给一个新的证明Eickmeyer和Kawarabayashi的结果，示出的继任者不变的一阶公式的模型检验问题是固定参数听话的排除拓扑未成年人的图类。
On the Complexity of the Positive Semidefinite Zero Forcing Number
图的正迫零数（英语：Positive zero forcing number）是一个由非传统类型的图着色所产生的图参数，并且与更传统的迫零版本有关。我们建立了迫零和快速混合搜索之间的关系，这意味着迫零问题的一些NP-完全性结果。对于弦图，关于正迫零和团覆盖之间的关系有很多了解。基于最优树覆盖和森林覆盖的构造，我们提出了一个线性时间算法来计算弦图的正迫零数。我们还证明了它是NP-完全的，以确定是否有一个图的正迫零集的一个附加的性质。
Faculty of Information Technology, Czech Technical University in Prague（布拉格捷克理工大学信息技术学院）；Computer Science Institute of Charles University, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院计算机科学研究所）
