Graphs excluding any fixed H as a d-fat minor admit balanced separators coverable by O(n^{1/2+ε}) radius-r balls, with a poly-time algorithm to find the separator or the fat model.
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Coarse Balanced Separators in Fat-Minor-Free Graphs
Graphs excluding any fixed H as a d-fat minor admit balanced separators coverable by O(n^{1/2+ε}) radius-r balls, with a poly-time algorithm to find the separator or the fat model.