# On fractional fragility rates of graph classes

## TL;DR

This paper investigates probabilistic vertex subsets in graphs with bounded inclusion probability, demonstrating how such sets can significantly fragment planar graphs into small components or low treedepth structures, with bounds close to optimal.

## Contribution

It introduces new probabilistic methods for graph fragmentation, establishing bounds on component sizes and treedepth for planar graphs with controlled vertex inclusion probabilities.

## Key findings

- Existence of probability distributions that fragment planar graphs into small components.
- Bounds on component size related to maximum degree and parameter a.
- Nearly-matching lower bounds for these fragmentation properties.

## 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.