The separation profile of any δ-hyperbolic planar or apex-minor-free graph grows at most as C log n, answering Benjamini–Schramm–Timár affirmatively.
Separation Number and Treewidth, Revisited
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We give a constructive proof of the fact that the treewidth of a graph $G$ is bounded by a linear function of the separation number of $G$.
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Separation profiles of hyperbolic planar and apex-minor-free graphs
The separation profile of any δ-hyperbolic planar or apex-minor-free graph grows at most as C log n, answering Benjamini–Schramm–Timár affirmatively.