Forbidding anticomplete planar minors: Induced Erdős--Pósa property and Maximum Independent Set in QP

The Erdős–Pósa theorem asserts that every graph $G$ with no $k$ disjoint cycles contains a set $X$ of $f(k)$ vertices such that $G\setminus X$ has no cycle. Robertson and Seymour showed that this Erdős–Pósa property also holds for $H$-minor models of any planar graph $H$. Equivalently, if $G$ has no $k$ minor models of $H$ pairwise at distance at least 1 (i.e. disjoint), then one can remove $f(k,H)$ balls of radius 0 (i.e. vertices) to make the graph $H$-minor free. We show that this coarse graph theory point of view generalizes to distance at least 2 versus radius 1 balls, yielding the induced Erdős–Pósa property for planar minors. Namely, every graph $G$ which does not contain $k$ pairwise non-adjacent minor models of a planar graph $H$ (we say that $G$ is $kH$-free) can be made $H$-minor free by removing $f(k,H)$ neighborhoods. The proof relies on the fact that sparse $kH$-free graphs have linearly many independent large protrusions. The same method gives that sparse $kH$-free graphs can be made $H$-minor free by deleting $O(\log n)$ vertices (and thus have logarithmic tree-width). This gives a quasi-polynomial algorithm for the Maximum Independent Set problem for $kH$-free graphs.