On Fractional Fragility Rates of Graph Classes

Zdeněk Dvořák, Jean-Sébastien Sereni
2020 Electronic Journal of Combinatorics  
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 for any set $X$ in the support of the distribution, the graph $G-X$ has component-size at most
more » ... \Delta(G)-1)^{a+O(\sqrt{a})}$, or treedepth at most $O(a^3\log_2(a))$. We also provide nearly-matching lower bounds.
doi:10.37236/8909 fatcat:cmcffslxnndfdene7px3ktey5i