A coarse Menger's Theorem for planar and bounded genus graphs
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866910209816920064 |
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| author | Blažej, Václav Pilipczuk, Michał Protopapas, Evangelos |
| author_facet | Blažej, Václav Pilipczuk, Michał Protopapas, Evangelos |
| contents | 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 $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11112 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A coarse Menger's Theorem for planar and bounded genus graphs Blažej, Václav Pilipczuk, Michał Protopapas, Evangelos Combinatorics Discrete Mathematics 05C10, 05C12, 05C38, 05C83, 68R10 G.2.1; G.2.2 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 $T$. |
| title | A coarse Menger's Theorem for planar and bounded genus graphs |
| topic | Combinatorics Discrete Mathematics 05C10, 05C12, 05C38, 05C83, 68R10 G.2.1; G.2.2 |
| url | https://arxiv.org/abs/2605.11112 |