# Hitting minors on bounded treewidth graphs. I. General upper bounds

## TL;DR

This paper investigates the parameterized complexity of hitting minors in graphs with bounded treewidth, establishing tight bounds on the running time based on the structure of the forbidden minors and the graph's properties.

## Contribution

It provides tight upper bounds on the complexity of ${\

## Key findings

- For general ${\
- For ${\
- For planar or surface-embedded graphs, the complexity is significantly reduced.

## Abstract

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem consists in, given a graph $G$ and an integer $k$, deciding whether there exists $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a minor. We are interested in the parameterized complexity of ${\cal F}$-M-DELETION when the parameter is the treewidth of $G$, denoted by $tw$. Our objective is to determine, for a fixed ${\cal F}$, the smallest function $f_{{\cal F}}$ such that {${\cal F}$-M-DELETION can be solved in time $f_{{\cal F}}(tw) \cdot n^{O(1)}$ on $n$-vertex graphs. We prove that $f_{{\cal F}}(tw) = 2^{2^{O(tw \cdot\log tw)}}$ for every collection ${\cal F}$, that $f_{{\cal F}}(tw) = 2^{O(tw \cdot\log tw)}$ if ${\cal F}$ contains a planar graph, and that $f_{{\cal F}}(tw) = 2^{O(tw)}$ if in addition the input graph $G$ is planar or embedded in a surface. We also consider the version of the problem where the graphs in ${\cal F}$ are forbidden as topological minors, called ${\cal F}$-TM-DELETION. We prove similar results for this problem, except that in the last two algorithms, instead of requiring ${\cal F}$ to contain a planar graph, we need it to contain a subcubic planar graph. This is the first of a series of articles on this topic.