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

2026-05-11

AI Summary

This work addresses a coarse-grained Menger-type connectivity problem for graphs embeddable on fixed surfaces (such as the plane or bounded-genus surfaces): when there are no $k$ pairwise vertex-disjoint $S$–$T$ paths with mutual distance greater than $d$, does there always exist a vertex set $X$ of bounded size such that every $S$–$T$ path lies within distance at most $d$ of some vertex in $X$? By integrating structural theorems from the theory of “colored” graph minors with topological properties of surface-embedded graphs, the authors establish the first such coarse Menger theorem for this graph class. Their result partially resolves an open problem posed by Nguyen, Scott, and Seymour and shows that the required separator $X$ has size bounded by a function $f(d,k)$ depending only on $d$, $k$, and the surface $\Sigma$.

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 $Σ$ 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 $Σ$-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 $T$.

Problem

• Menger's Theorem
• planar graphs
• bounded genus graphs
• vertex-disjoint paths
• coarse variant

Innovation

• coarse Menger's Theorem
• planar graphs
• bounded genus graphs
• colorful graph minor theory
• distance-disjoint paths