Coarse balanced separators versus coarse tree decompositions

Establish that for every pair of positive integers k and r there exist integers ℓ and d, depending only on k and r, such that every graph in which every nonnegative vertex-weight function admits a balanced separator coverable by k balls of radius r has a tree decomposition whose every bag is coverable by ℓ balls of radius d.

Background

The central unresolved problem concerns whether the classical functional equivalence between treewidth and balanced separator number persists in coarse graph theory. A set is considered coarse-small when it can be covered by a bounded number of bounded-radius balls. The conjecture asks whether uniformly coarse-small balanced separators for all vertex weightings force a tree decomposition with uniformly coarse-small bags, with the covering parameters independent of the size of the graph.

The paper proves the conjecture under a bounded-doubling-dimension assumption and establishes weaker results in general graphs, but explicitly leaves the full conjecture unresolved; even the case of radius parameter r=1 remains open.

References

For all $k,r\in N$ there exist $\ell,d\in N$ such that the following holds. Suppose $G$ is a graph such that every weight function $\mu\colon V(G)\to R_{ 0}$ admits a balanced separator that is $(k,r)$-coverable. Then $G$ admits a tree decomposition whose every bag is $(\ell,d)$-coverable.\end{conjecture}

The converse implication is easy: if $G$ admits a tree decomposition $T$ whose bags are $(\ell,d)$-coverable and $\mu$ is a weight function on $G$, then again there is a bag of $T$ that is a balanced separator for $\mu$, hence $\mu$ has an $(\ell,d)$-coverable balanced separator.

We do not resolve \cref{con:main} in this work; in fact, even the resolution of case $r=1$ would be very interesting.

— On coarse tree decompositions and coarse balanced separators  (2502.20182 - Abrishami et al., 27 Feb 2025) in Conjecture 1, Section 1, Introduction