A class of graphs is structurally nowhere dense if it can be constructed from a nowhere dense class by a first-order transduction. Structurally nowhere dense classes vastly generalize nowhere dense classes and constitute important examples of monadically stable classes. We show that the first-order model checking problem is fixed-parameter tractable on every structurally nowhere dense class of graphs. Our result builds on a recently developed game-theoretic characterization of monadically stable graph classes. As a second key ingredient of independent interest, we provide a polynomial-time algorithm for approximating weak neighborhood covers (on general graphs). We combine the two tools into a recursive locality-based model checking algorithm. This algorithm is efficient on every monadically stable graph class admitting flip-closed sparse weak neighborhood covers, where flip-closure is a mild additional assumption. Thereby, establishing efficient first-order model checking on monadically stable classes is reduced to proving the existence of flip-closed sparse weak neighborhood covers on these classes - a purely combinatorial problem. We complete the picture by proving the existence of the desired covers for structurally nowhere dense classes: we show that every structurally nowhere dense class can be sparsified by contracting local sets of vertices, enabling us to lift the existence of covers from sparse classes.
翻译:类图如果可以通过一阶转换,从一个不高密的舱层中建造出,则在结构上是密不可分的。 结构上不密的舱层极密, 粗密的舱层极密, 粗密的舱层非常普遍, 并且构成月度稳定级的重要例子。 我们显示, 第一阶模型检查问题是固定的参数, 每一个结构上不稠密的图表层都可绘制。 我们的结果基于最近开发的对月度稳定的图表层的游戏理论特征。 作为独立兴趣的第二个关键成分, 我们为相近的弱小区盖提供了一种多元时间算法( 在一般图形中)。 我们把两种工具合并成一个循环性的基于地点的模型检查算法。 我们的算法对每个单调稳定的图表层结构上都是高效的, 承认翻动的稀疏弱的街区层覆盖, 翻动的偏移是一个简单的假设。 因此, 在单调稳定的舱上建立高效的一阶模型, 以证明这些教室上存在翻动的稀弱的棚子覆盖―― 一个纯粹的梳子问题。 我们通过证明结构上低层层层层层的层的升级而完成图片, 显示每个结构层的升级的升级的升级的升级的升级的升级的升级的等级。