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].

翻译：设$A$和$B$是图$G$中的顶点集。门格尔定理指出：对于每个正整数$k$，要么存在$k$条顶点不交的$A$-$B$路径，要么存在最多$k-1$个顶点构成的集合将$A$与$B$分离。令$\Delta$为$G$的最大度。我们证明存在函数$f(\Delta) = (\Delta+1)^{\Delta^2+1}$，使得对于每个正整数$k$，要么存在$k$条顶点不交且两两反完备（anticomplete）的$A$-$B$路径，要么存在最多$k \cdot f(\Delta)$个顶点构成的集合将$A$与$B$分离。我们还证明该结果可以从有界度图推广到排除某个拓扑子式的图。在否定方面，我们证明当退化度（degeneracy）为2且围长（girth）任意大时，即使$k=2$，也不存在此类关系。Hendrey、Norin、Steiner和Turcotte [arXiv:2309.07905] 独立且同时得到了类似结果。