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arXiv · 1907.12634

On fractional fragility rates of graph classes

Zdeněk Dvořák·Jean-Sébastien Sereni
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Abstract

We consider, for every positive integer $a$, probability distributions on subsets of vertices of a graph with the property that every vertex belongs to the random set sampled from this distribution with probability at most $1/a$. Among other results, we prove that for every positive integer~$a$ and every planar graph $G$, there exists such a probability distribution with the additional property that deleting the random set creates a graph with component-size at most $(\Delta(G)-1)^{a+O(\sqrt{a})}$, or a graph with treedepth at most $O(a^3\log_2(a))$. We also provide nearly-matching lower bounds.

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Zdeněk Dvořák, Jean-Sébastien Sereni. 2019-07-29. On fractional fragility rates of graph classes. https://arxiv.org/abs/1907.12634

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