排名宽度符合稳定性
Rankwidth meets stability
摘要：
我们研究了两个概念，被良好的结构类图的灵感来自经典模型理论。一类图$C$是单子稳定的，如果不可能用固定的一阶公式从$C$定义顶点着色图的任意长的线性阶.类似地，一元依赖对应于不可能以这种方式定义所有图。单子稳定的图类的例子是无处密集类，它提供了一个强大的稀疏性理论。单子依赖类的例子是有界秩宽（或等价地，有界秩宽）的类，它可以被看作是有界树宽类的密集模拟。因此，一元稳定性和一元依赖性扩展了经典的结构概念，通过在更广泛的模型理论背景下查看它们。我们探索这一新兴的理论证明如下：-一类图$C$是一个一阶转换一类有界树宽当且仅当$C$有界秩宽和一个稳定的边缘关系（即从$C$排除一些半图作为半诱导子图）。- 如果一类图$C$是单依赖的且不是单稳定的，则$C$实际上有一个不稳定的边关系.因此，我们证明了除了某些半图作为半导出子图之外，具有有界秩宽的类是线性$χ$ -有界的。我们的证明是有效的，并导致多项式时间算法。
英文摘要：
We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs $C$ is monadically stable if it is impossible to define arbitrarily long linear orders in vertex-colored graphs from $C$ using a fixed first-order formula. Similarly, monadic dependence corresponds to the impossibility of defining all graphs in this way. Examples of monadically stable graph classes are nowhere dense classes, which provide a robust theory of sparsity. Examples of monadically dependent classes are classes of bounded rankwidth (or equivalently, bounded cliquewidth), which can be seen as a dense analog of classes of bounded treewidth. Thus, monadic stability and monadic dependence extend classical structural notions for graphs by viewing them in a wider, model-theoretical context. We explore this emerging theory by proving the following: - A class of graphs $C$ is a first-order transduction of a class with bounded treewidth if and only if $C$ has bounded rankwidth and a stable edge relation (i.e. graphs from $C$ exclude some half-graph as a semi-induced subgraph). - If a class of graphs $C$ is monadically dependent and not monadically stable, then $C$ has in fact an unstable edge relation. As a consequence, we show that classes with bounded rankwidth excluding some half-graph as a semi-induced subgraph are linearly $χ$-bounded. Our proofs are effective and lead to polynomial time algorithms.
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