Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths – a result that shows a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes. The structural results we obtain are the following. (1) The number of unlabeled graphs of order $n$ with linear rank-width at most $r$ is at most ${\left[(2^r+1)(r+1)!2^{\left(\genfrac{}{}{0ex}{}{r}{2}\right)}{3}^{r+1}\right]}^n$ (2) Graphs with linear rankwidth at most $r$ are linearly $\chi$-bounded. Actually, they have bounded $c$-chromatic number, meaning that they can be colored with $f\left(r\right)$ colors, each color inducing a cograph. (3) To the contrary, based on a Ramsey-like argument, we prove for every proper hereditary family $ℱ$ of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in $ℱ$.

From the model theoretical side we obtain the following results: (1) A direct short proof that graphs with linear rankwidth at most $r$ are first-order transductions of linear orders. This result could also be derived from Colcombet’s theorem on first-order transduction of linear orders and the equivalence of linear rankwidth with linear cliquewidth. (2) For a class $\mathcal{C}$ with bounded linear rankwidth the following conditions are equivalent: (a) $\mathcal{C}$ is stable, (b) $\mathcal{C}$ excludes some half-graph as a semi-induced subgraph, (c) $\mathcal{C}$ is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers.

具有限定秩宽度的类是树的MSO转换，具有限定线性秩宽度的类是路径的MSO转换-结果表明，从结构图理论的角度和从结构图理论的角度来看，这些图类的属性之间有着密切的联系。有限模型理论的观点。我们对具有有限线性秩宽度的类都采取了两种观点，并证明了这些类的结构和模型理论性质。我们获得的结构结果如下。（1）顺序的未标记图的数量$ñ$ 最多具有线性秩宽度 $\mathrm{[R}$ 最多 ${\left[（2^{\mathrm{[R}}+\mathrm{1个}）（\mathrm{[R}+\mathrm{1个}）！2^{\left(\genfrac{}{}{0ex}{}{\mathrm{[R}}{2}\right)}{3}^{\mathrm{[R}+\mathrm{1个}}\right]}^{ñ}$ （2）最多具有线性秩宽度的图 $\mathrm{[R}$ 线性 $\chi$界。实际上，他们已经$C$色数，表示它们可以用 $F（\mathrm{[R}）$的颜色，每种颜色都会引起一个笔势。（3）相反，基于类似Ramsey的论点，我们证明了每个适当的世袭家族$ℱ$ 的图（如cograph）中，存在一个具有有限的rankwidth的类，该类不具有其中的图可以由一定数量的颜色着色的属性，每个颜色都在 $ℱ$。

从模型理论的角度，我们得到以下结果：（1）一个直接的简短证明，即最多具有线性秩宽度的图 $\mathrm{[R}$是线性阶的一阶转换。该结果也可以从关于线性阶的一阶转导的Colcombet定理以及线性秩宽度与线性团宽度的等价关系中得出。（2）上课$\mathcal{C}$ 在具有有限线性rankwidth的情况下，以下条件是等效的：（a） $\mathcal{C}$ 稳定，（b） $\mathcal{C}$ 不包括一些半图作为半诱导子图，（c） $\mathcal{C}$是具有有限路径宽度的类的一阶转换。这些结果为研究低线性秩宽覆盖率的课程开辟了前景。