For every integer d≥2, the paper constructs graphs of degree-d polynomial growth whose balanced separators are too large by a logarithmic factor to fit in the conjectured product structure.
Polynomial-time approximation schemes for induced subgraph problems on fractionally tree-independence-number-fragile graphs
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abstract
We investigate a relaxation of the notion of fractional treewidth-fragility, namely fractional tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for meta-problems such as finding a maximum-weight sparse induced subgraph satisfying a given $\mathsf{CMSO}_2$ formula on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth, and its applications to exact subexponential-time algorithms.
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Subdivided expanders and counterexamples to the Tree Product Conjecture
For every integer d≥2, the paper constructs graphs of degree-d polynomial growth whose balanced separators are too large by a logarithmic factor to fit in the conjectured product structure.