Coarse Balanced Separators in Fat-Minor-Free Graphs
This result extends the fundamental balanced separator theorem to the coarse setting of fat-minor-free graphs, which is relevant for graph theory and algorithm design.
The paper provides a coarse analogue of the classic balanced separator theorem for graphs excluding a fixed minor, proving that any n-vertex graph excluding H as a d-fat minor has a balanced separator that can be covered by O(n^{1/2+ε}) balls of constant radius. The result extends to weighted settings and yields a randomized polynomial-time algorithm to find such a separator or a fat minor model.
Fat minors are a coarse analogue of graph minors where the subgraphs modeling vertices and edges of the embedded graph are required to be distant from each other, instead of just being disjoint. In this paper, we give a coarse analogue of the classic theorem that an $n$-vertex graph excluding a fixed minor admits a balanced separator of size $O(\sqrt{n})$. Specifically, we prove that for every integer $d$, real $\varepsilon>0$, and graph $H$, there exist constants $c$ and $r$ such that every $n$-vertex graph $G$ excluding $H$ as a $d$-fat minor admits a set $S \subseteq V(G)$ that is a balanced separator of $G$ and can be covered by $c n^{\frac{1}{2}+\varepsilon}$ balls of radius $r$ in $G$. Our proof also works in the weighted setting where the balance of the separator is measured with respect to any weight function on the vertices, and is effective: we obtain a randomized polynomial-time algorithm to compute either such a balanced separator, or a $d$-fat model of $H$ in $G$.