AI 中文总结
研究生成路由设置中无关顶点技术的组合限制，确定注释图新参数\(\mathsf{depth}_2\)，证明对于红色子式封闭的注释图类，该技术适用于生成不相交路径当且仅当类具有有界\(\mathsf{depth}_2\)，给出算法时间及匹配下界。
AI 中文摘要
无关顶点技术是算法图论的基石之一，是Robertson和Seymour的不相交路径算法及许多算法图子式理论的基础。我们表明，在生成路由设置中，该技术存在精确的组合限制。与经典路由问题不同，生成路由不由特殊顶点的数量决定，而是由它们在图中的分布方式决定。输入是三元组\((G,R,\mathcal{T})\)，目标是确定\(G\)是否包含一组内部不相交的路径，连接\(\mathcal{T}\)中的对，使得路径的并集覆盖集合\(R\)。我们确定了注释图的一个新结构参数\(\mathsf{depth}_2\)，它精确地衡量了这种现象。我们的主要结果是一个完整的组合二分法：对于每个红色子式封闭的注释图类，无关顶点技术适用于生成不相交路径当且仅当该类具有有界的\(\mathsf{depth}_2\)。因此，\(\mathsf{depth}_2\)形成了Robertson - Seymour范式幸存和失效的类之间的精确结构边界。我们的证明结合了有界\(\mathsf{depth}_2\)的注释图的新局部结构定理和著名的重要链接定理的生成类似物。所得算法在时间\(2^{2^{\mathbf{poly}(k + d)}}\cdot n^2\)内解决生成不相交路径问题，其中\(d\)是输入实例的\(\mathsf{depth}_2\)。我们提供了匹配的下界，表明即使在平面图上，超过有界\(\mathsf{depth}_2\)也不存在无关顶点规则。特别是，\(\mathsf{depth}_2\)是生成约束下无关顶点技术的精确组合障碍。
英文摘要
The Irrelevant Vertex Technique is one of the cornerstones of algorithmic graph theory, underlying Robertson and Seymour's algorithm for \textsc{Disjoint Paths} and much of the algorithmic Graph Minors theory. We show that, in the setting of spanning routing, this technique exhibits an exact combinatorial limitation. Unlike classical routing problems, spanning routing is not governed by the number of distinguished vertices but by the way they are distributed throughout the graph. The input is a triple $(G,R,\mathcal{T})$ where $(G,R)$ is an annotated graph and $\mathcal{T}$ is a set of terminal pairs. The goal is to determine if $G$ contains a family of internally disjoint paths connecting the pairs in $\mathcal{T}$ such that the union of the paths spans the set $R$. We identify a new structural parameter of annotated graphs, called $\mathsf{depth}_2$, that measures precisely this phenomenon. Our main result is a complete combinatorial dichotomy: for every red-minor-closed class of annotated graphs, the Irrelevant Vertex Technique applies to \textsc{Spanning Disjoint Paths} \textsl{if and only if} the class has bounded $\mathsf{depth}_2$. Thus $\mathsf{depth}_2$ forms the exact structural boundary between classes where the Robertson-Seymour paradigm survives and those where it breaks down. Our proof combines a new local structure theorem for annotated graphs of bounded $\mathsf{depth}_2$ with a spanning analogue of the celebrated Vital Linkage Theorem. The resulting algorithm solves \textsc{Spanning Disjoint Paths} in time $2^{2^{\mathbf{poly}(k+d)}}\cdot n^2$ where $d$ is the $\mathsf{depth}_2$ of the input instance. We provide matching lower bounds showing that beyond bounded $\mathsf{depth}_2$ no irrelevant-vertex rule can exist, even on planar graphs. In particular, $\mathsf{depth}_2$ is the exact combinatorial barrier for the Irrelevant Vertex Technique under spanning constraints.
