Connections between structural graph theory and finite model theory recently gained a lot of attention. In this setting, many interesting questions remain on the properties of dependent (NIP) hereditary classes of graphs, in particular related to first-order transductions. In this paper, we study modelizations (which are strong forms of transduction pairings) of classes of graphs by classes of structures. In particular, we consider models obtained by coupling a partial order and a colored graph (thus forming a partially ordered colored graph). Motivated by Simon's decomposition theorem of dependent types into a stable part and a distal (order-like) part, we conjecture that every dependent hereditary class of graphs admits a modelization in a monadically dependent coupling of a class of posets with bounded treewidth cover graphs and a monadically stable class of colored graphs. In this paper, we consider the first non-trivial case (classes with bounded linear cliquewidth) and prove that the conjecture holds in a strong form, the model class being a monadically dependent coupling of a class of disjoint unions of chains and a class of colored graphs with bounded pathwidth. We extend our study to classes that admit bounded-size bounded linear cliquewidth decompositions and prove that they have a modelization in a monadically dependent coupling of a class of disjoint unions of chains and a class of colored graphs with bounded expansion, the model class also admitting bounded-size bounded linear cliquewidth decompositions.

翻译：结构图论与有限模型论之间的联系近来备受关注。在此背景下，依赖（NIP）遗传图类的性质仍存在许多有趣问题，尤其涉及一阶转换。本文研究图类通过结构类实现的模型化（即转换配对的强化形式）。特别地，我们考虑通过耦合偏序与着色图（从而形成偏序着色图）获得的模型。受西蒙将依赖类型分解为稳定部分与远端（类序）部分的定理启发，我们提出猜想：每个依赖遗传图类均可在具有有界树宽覆盖图的偏序类与单子依赖着色图类的单子依赖耦合中实现模型化。本文研究首个非平凡情形（具有有界线性团宽的图类），并以强形式证明该猜想成立——模型类为链的不交并类与具有有界路径宽的着色图类的单子依赖耦合。我们将研究拓展至具有有界规模有界线性团宽分解的图类，证明其可在链的不交并类与具有有界扩张的着色图类的单子依赖耦合中实现模型化，且该模型类同样允许有界规模有界线性团宽分解。