A coarse Menger's Theorem for planar and bounded genus graphs

TL;DR

This paper extends Menger's Theorem to planar and bounded genus graphs by establishing a relation between the absence of large-distance disjoint paths and small vertex sets covering all paths.

Contribution

It proves a coarse variant of Menger's Theorem for graphs embeddable on surfaces, addressing a question left open in prior research.

Findings

For every surface, a function f bounds the size of vertex sets covering all S-T paths within distance d.

If no k S-T paths are pairwise at distance greater than d, then a small vertex set exists to cover all S-T paths within distance d.

The result partially answers a question by Nguyen, Scott, and Seymour about path structures in surface-embeddable graphs.

Abstract

Menger's Theorem is a fundamental result in graph theory. It states that if in a graph $G$ with distinguished sets of terminal vertices $S$ and $T$ there are no $k$ pairwise vertex-disjoint $S$-$T$ paths, then there is a set of less than $k$ vertices that intersects every $S$-$T$ path. In this work, we give a coarse variant of this result for planar and bounded genus graphs. Precisely, we prove that for every surface $\Sigma$ there is a function $f\colon \mathbb{N}\times \mathbb{N}\to \mathbb{N}$ such that for every pair of integers $d,k\in \mathbb{N}$ and a $\Sigma$-embeddable graph $G$ with distinguished sets of terminal vertices $S$ and $T$, if $G$ does not contain a family of $k$ $S$-$T$ paths that are pairwise at distance larger than $d$, then there is a set $X$ consisting of at most $f(d,k)$ vertices of $G$ such that every $S$-$T$ path is at distance at most $d$ from a vertex of $X$. This partially answers questions of Nguyen, Scott, and Seymour [arXiv:2508.14332], who proved that such a result cannot hold in general graphs. A key ingredient of our proof is a structure theorem from the developing ''colorful'' graph minor theory, where the focus is on studying the structure in a graph relative to some fixed subsets of annotated vertices. In our case, these annotated vertices are $S$ and $T$.