We provide proofs certifying that the structure theorem for vertex sets of bounded bidimensionality holds with polynomial bounds. The bidimensionality of vertex sets is a common generalisation of both treewidth and the face-cover-number of vertex sets in planar graphs. As such, it plays a crucial role in extensions of Courcelle's Theorem to $H$-minor-free graphs. Recently, bidimensionality and similar parameters have emerged as key for extensions of known parameterized algorithms for problems defined on a terminal set $R$. A prominent example for such a problem is Steiner Tree, which admits efficient algorithms on planar graphs whenever $R$ can be covered with few faces. Key to the algorithmic applications of bidimensionality is a structure theorem that explains how a graph $G$ can be decomposed into pieces where the behaviour of $R$ is highly controlled. One may see this structure theorem as a rooted analogue of Robertson and Seymour's celebrated Grid Theorem. Combining recent advances in obtaining polynomial bounds in the Graph Minors framework with new techniques for handling annotated vertex sets, we show that all parameters in the structure theorem above admit polynomial bounds. As an application, we also provide a sketch showing how our techniques imply polynomial bounds for the structure theorem for graphs excluding an apex minor.

翻译：我们提供了证明，确认有界二维度的顶点集结构定理具有多项式界。顶点集的二维度是平面图中树宽与面覆盖数的共同推广。因此，它在将库尔塞勒定理推广至$H$-次要自由图的过程中起着关键作用。近年来，二维度及类似参数已成为扩展针对定义在终端集$R$上的问题的已知参数化算法的关键。此类问题的一个突出例子是斯坦纳树问题，当$R$能被少数面覆盖时，该问题在平面图上存在高效算法。二维度算法应用的关键在于一个结构定理，该定理解释了图$G$如何被分解为若干部分，其中$R$的行为受到高度控制。可将此结构定理视为罗伯逊-西摩著名网格定理的有根类比。通过将图次要框架中获得多项式界的最新进展与处理带标注顶点集的新技术相结合，我们证明了上述结构定理中的所有参数均具有多项式界。作为应用，我们还提供了一个概要，说明我们的技术如何为排除尖点次要图的结构定理蕴含多项式界。