Long induced paths in minor-closed graph classes and beyond
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In this paper we show that every graph of pathwidth less than k that has a path of order n also has an induced path of order at least 1/3 n^1/k. This is an exponential improvement and a generalization of the polylogarithmic bounds obtained by Esperet, Lemoine and Maffray (2016) for interval graphs of bounded clique number. We complement this result with an upper-bound. This result is then used to prove the two following generalizations: - every graph of treewidth less than k that has a path of order n contains an induced path of order at least 1/4 (log n)^1/k; - for every non-trivial graph class that is closed under topological minors there is a constant d ∈ (0,1) such that every graph from this class that has a path of order n contains an induced path of order at least (log n)^d. We also describe consequences of these results beyond graph classes that are closed under topological minors.
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