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.
@article{10_37236_8909,
author = {Zden\v{e}k Dvo\v{r}\'ak and Jean-S\'ebastien Sereni},
title = {On fractional fragility rates of graph classes},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {4},
doi = {10.37236/8909},
zbl = {1471.60011},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8909/}
}
TY - JOUR
AU - Zdeněk Dvořák
AU - Jean-Sébastien Sereni
TI - On fractional fragility rates of graph classes
JO - The electronic journal of combinatorics
PY - 2020
VL - 27
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/8909/
DO - 10.37236/8909
ID - 10_37236_8909
ER -
%0 Journal Article
%A Zdeněk Dvořák
%A Jean-Sébastien Sereni
%T On fractional fragility rates of graph classes
%J The electronic journal of combinatorics
%D 2020
%V 27
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/8909/
%R 10.37236/8909
%F 10_37236_8909
Zdeněk Dvořák; Jean-Sébastien Sereni. On fractional fragility rates of graph classes. The electronic journal of combinatorics, Tome 27 (2020) no. 4. doi: 10.37236/8909
