A class of structures is monadically dependent if one cannot interpret all graphs in colored expansions from the class using a fixed first-order formula. A tree-ordered $σ$-structure is the expansion of a $σ$-structure with a tree-order. A tree-ordered $σ$-structure is weakly sparse if the Gaifman graph of its $σ$-reduct excludes some biclique (of a given fixed size) as a subgraph. Tree-ordered weakly sparse graphs are commonly used as tree-models (for example for classes with bounded shrubdepth, structurally bounded expansion, bounded cliquewidth, or bounded twin-width), motivating their study on their own. In this paper, we consider several constructions on tree-ordered structures, such as tree-ordered variants of the Gaifman graph and of the incidence graph, induced and non-induced tree-ordered minors, and generalized fundamental graphs. We provide characterizations of monadically dependent classes of tree-ordered weakly sparse $σ$-structures based on each of these constructions, some of them establishing unexpected bridges with sparsity theory. As an application, we prove that a class of tree-ordered weakly sparse structures is monadically dependent if and only if its sparsification is nowhere-dense. Moreover, the sparsification transduction translates boundedness of clique-width and linear clique-width into boundedness of tree-width and path-width. We also prove that first-order model checking is not fixed parameter tractable on independent hereditary classes of tree-ordered weakly sparse graphs (assuming $AW[*]\neq FPT$) and give what we believe is the first model-theoretical characterization of classes of graphs excluding a minor, thus opening a new perspective of structural graph theory.

翻译：一个结构类是单依赖的，如果无法通过固定的一个一阶公式在着色扩张中解释所有图。树序$σ$-结构是$σ$-结构通过树序扩张得到的结构。一个树序$σ$-结构是弱稀疏的，如果其$σ$-归约的盖弗曼图排除了某个二分团（给定固定大小）作为子图。树序弱稀疏图通常被用作树模型（例如用于具有有界灌木深度、结构有界扩张、有界团宽度或有界孪生宽度的类），这促使了对它们自身的研究。在本文中，我们考虑了树序结构上的几种构造，例如盖弗曼图和关联图的树序变体、诱导和非诱导的树序子式，以及广义基本图。我们基于这些构造中的每一个，提供了树序弱稀疏$σ$-结构类的单依赖性的刻画，其中一些刻画在稀疏性理论中建立了意想不到的桥梁。作为一个应用，我们证明了一个树序弱稀疏结构类是单依赖的当且仅当其稀疏化是无处稠密的。此外，稀疏化转换将团宽度和线性团宽度的有界性转化为树宽度和路径宽度的有界性。我们还证明了在树序弱稀疏图的独立遗传类上，一阶模型检测不是固定参数可处理的（假设$AW[*]\neq FPT$），并给出了我们所认为的第一个排除子式的图类的模型论刻画，从而为结构图论开辟了新的视角。