The Graph Minors Series of Robertson and Seymour forms the foundation of algorithmic structural graph theory, yielding fixed-parameter algorithms for problems such as Disjoint Paths, Rooted Minor Checking, and Folio. A key ingredient behind the fixed-parameter tractability of the $k$-Disjoint Paths problem is the irrelevant-vertex technique. This machinery is governed by the Vital Linkage Theorem and the so-called Linkage Function $\ell$. However, despite its foundational role, the best known bounds on the Linkage Function are enormous and are only implicitly understood. The quantitative bounds behind these results have traditionally been so large that the resulting algorithms are regarded as "galactic". Our main result is a general irrelevant-vertex theorem for a common generalisation of $k$-Disjoint Paths and Rooted Minor Checking for graphs of size at most $d,$ commonly called the $(k,d)$-Folio problem. Specifically, we show that for any graph $G$ in which the $k$ terminals are chosen from some set $R,$ if the treewidth of $G$ exceeds $β(k,b,d)\in$ $2^{{\bf poly}(b + d)}$ $\cdot {\bf poly}(k)$ then we can locate an irrelevant vertex for the $(k,d)$-Folio problem. Here, the quantity $b$ is the bidimensionality of $R,$ that is, the largest $b$ for which a $(b\times b)$-grid minor in $G$ can be rooted on $R$. Thus, the exponential component of the irrelevant-vertex threshold is driven by the bound on the bidimensionality, rather than by the number of terminals, and we argue that this dependence is essentially optimal up to polynomial factors. As a consequence, the Linkage Function satisfies $\ell(k) \in 2^{{\bf poly}(k)}$. Beyond its structural significance, our result yields improved parameter dependencies for algorithms for Disjoint Paths and Rooted Minor Checking}, and provides a quantitative improvement for a broad range of graph-minor-based algorithmic frameworks.

翻译：罗伯逊与西摩的图子系列构成了算法结构图论的基础，为不交路径、有根子式检测及文集问题等提供了固定参数算法。k-不交路径问题的固定参数可解性依赖于不相关顶点技术，该技术由关键链接定理及链接函数ℓ控制。然而，尽管其基础性作用，链接函数的最佳已知界限极其庞大且仅被隐式理解。这些结果背后的数量界限传统上如此之大，以至于生成的算法被视为"天文级"。我们的主要成果是针对k-不交路径与有根子式检测（限定图规模不超过d，通常称为(k,d)-文集问题）的广义版本，提出通用的不相关顶点定理。具体而言，对于任意图G（其k个端点选自某集合R），若G的树宽超过β(k,b,d)∈2^{poly(b+d)}·poly(k)，则我们可定位(k,d)-文集问题的不相关顶点。其中b是R的二维性，即G中可扎根于R的(b×b)-网格子图的最大尺寸。因此，不相关顶点阈值的指数分量由二维性界限驱动，而非端点数量，且我们证明此依赖关系在多项式因子范围内实质上最优。作为推论，链接函数满足ℓ(k)∈2^{poly(k)}。除结构意义外，我们的结果为不交路径与有根子式检测算法带来改进的参数依赖性，并为基于图子式的广泛算法框架提供定量优化。