  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].
