Spanning Paths and Cycles: Structural Limitations of the Irrelevant Vertex Technique

2026-07-10

AI Summary

This study investigates the applicability limits of irrelevant-vertex techniques for the disjoint paths problem with crossing constraints. Focusing on the distribution of terminal vertices in labeled graphs, the authors introduce a new structural parameter, depth₂, and prove that it precisely characterizes the effectiveness of irrelevant-vertex rules: when depth₂ is bounded, a fixed-parameter algorithm exists with running time $2^{2^{\text{poly}(k+d)}} \cdot n^2$; however, if depth₂ is unbounded, no such rule exists even on planar graphs. By integrating graph minor theory and structural graph theory, the work establishes a local structure theorem for graph classes with bounded depth₂ and a key connectivity theorem for routed path variants, leading to novel irrelevant-vertex reduction rules and decomposition methods tailored to labeled graphs.

Abstract

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.

Problem

• Spanning Disjoint Paths
• Irrelevant Vertex Technique
• Graph Minors
• Structural Parameter
• depth_2

Innovation

• Irrelevant Vertex Technique
• Spanning Disjoint Paths
• depth_2
• Graph Minors
• Structural Parameter