On fractional fragility rates of graph classes

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 $(\Delta(G)-1)^{a+O(\sqrt{a})}$, or treedepth at most $O(a^3\log_2(a))$. We also provide nearly-matching lower bounds.