On Induced Versions of Menger's Theorem on Sparse Graphs

TL;DR

This paper extends Menger's theorem to sparse graphs, establishing bounds on vertex-disjoint paths and separators in graphs with bounded degree or excluding a topological minor, with some limitations in specific graph classes.

Contribution

It introduces a function bounding the size of separators for vertex-disjoint paths in bounded-degree and minor-excluding graphs, generalizing classical Menger's theorem.

Findings

Existence of a function f(Δ) bounding separators in bounded-degree graphs.

Generalization of the result to graphs excluding a topological minor.

Counterexamples in graphs with degeneracy 2 and large girth.

Abstract

Let $A$ and $B$ be sets of vertices in a graph $G$. Menger's theorem states that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of at most $k-1$ vertices. Let $\Delta$ be the maximum degree of $G$. We show that there exists a function $f(\Delta) = (\Delta+1)^{\Delta^2+1}$, so that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint and pairwise anticomplete paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of at most $k \cdot f(\Delta)$ vertices. We also show that the result can be generalized from bounded-degree graphs to graphs excluding a topological minor. On the negative side, we show that no such relation holds on graphs that have degeneracy 2 and arbitrarily large girth, even when $k = 2$. Similar results were obtained independently and concurrently by Hendrey, Norin, Steiner, and Turcotte [arXiv:2309.07905].