The Local Structure Theorem (LST) for Graph Minors roughly states that for every $H$-minor-free graph $G$ that contains a sufficiently large wall $W$, there is a small vertex subset $A,$ whose removal yields a graph that admits an "almost embedding" $δ$ on a surface $Σ$ on which $H$ does not embed. By almost embedding, we mean that there exists a hypergraph $\mathcal{H}$ whose vertex set is a subset of the vertex set of $G - A$ and an embedding of $\mathcal{H}$ on $Σ$ such that the drawing of each hyperedge of $\mathcal{H}$ corresponds to a cell of $δ,$ the boundary of each cell intersects only the vertices of the corresponding hyperedge, and all remaining vertices and edges of $G - A$ are drawn in the interior of cells. The cells corresponding to hyperedges of arity at least $4$, called vortices, are few in number and have small "depth", while "most" of the wall $W$ is disjoint from the vortices and is "grounded" in the embedding $δ$. Suppose that the subgraphs drawn inside each of the non-vortex cells are equipped with some finite index, i.e., each such cell is assigned a color from a finite set. We prove a version of the LST in which the set $C$ of colors assigned to the non-vortex cells exhibits "large" bidimensionality: $G - A$ contains a minor model of a large grid $Γ$ such that, for every color $α\in C$, the model of each vertex of $Γ$ contains the subgraph drawn within an $α$-colored cell. Moreover, $Γ$ can be chosen in a way that is "well-connected" to the original wall $W$.

翻译：局部结构定理（LST）关于图子式粗略而言指出：对于每个不含$H$作为子式的图$G$，若它包含一个足够大的墙$W$，则存在一个小顶点子集$A$，移除$A$后所得图在曲面$Σ$上允许一个“几乎嵌入”$δ$，且$H$无法嵌入到$Σ$上。所谓几乎嵌入，是指存在一个超图$\mathcal{H}$，其顶点集是$G-A$顶点集的子集，且$\mathcal{H}$可嵌入到$Σ$上，使得$\mathcal{H}$的每条超边画法对应$δ$中的一个胞腔，每个胞腔的边界仅与对应超边的顶点相交，而$G-A$的所有剩余顶点和边则画在胞腔内部。对应于arity至少为4的超边的胞腔（称为涡旋）数量少且“深度”小，而墙$W$的“大部分”与涡旋不相交，并在嵌入$δ$中“接地”。假设每个非涡旋胞腔内部所画的子图配备某种有限指标，即每个此类胞腔被赋予有限颜色集合中的一种颜色。我们证明一个版本的LST，其中非涡旋胞腔的颜色集合$C$展现出“大”双维性：$G-A$包含一个大网格$Γ$的子式模型，使得对于每种颜色$α∈C$，$Γ$每个顶点的模型都包含画在$α$色胞腔内的子图。此外，$Γ$可选择为与原墙$W$“良好连通”的方式。