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 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.
Highlights
Planar graphs “almost” have bounded treewidth, in the following sense: For every assignment of weights to vertices and for every positive integer a, it is possible to delete vertices of at most 1/a fraction of the total weight so that the resulting subgraph has treewidth at most 3a − 3
There exists a probability distribution on subsets of vertices whose complement induces a subgraph of treewidth at most 3a − 3, such that each vertex belongs to a set sampled from this distribution with probability at most 1/a
For ε > 0, we say that a probability distribution on the subsets of vertices of a graph G is ε-thin if for each vertex v, the probability that v belongs to a set sampled from this distribution is at most ε
Summary
Planar graphs “almost” have bounded treewidth, in the following sense: For every assignment of weights to vertices and for every positive integer a, it is possible to delete vertices of at most 1/a fraction of the total weight so that the resulting subgraph has treewidth at most 3a − 3. There exists a probability distribution on subsets of vertices whose complement induces a subgraph of treewidth at most 3a − 3, such that each vertex belongs to a set sampled from this distribution with probability at most 1/a This property is the key ingredient of a number of approximation algorithms for planar graphs [1]. Suppose that a class G of graphs is fractionally f -fragile at rate r and, that there exists an algorithm that for a graph G ∈ G and a positive integer a returns a set sampled from a (1/a)-thin probability distribution on Gf↓r(a) in polynomial time. Let us remark that in essentially all known cases of fractionally fragile classes, it is possible to find a probability distribution as described in Observation 1 with support of polynomial size, and the algorithm can be derandomized by trying all sets from the support rather than sampling one of them. As we will see below (Lemma 2), fractional f -fragility is equivalent to fractional f -colorability by 1 color, at a matching rate
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