Hitting minors on bounded treewidth graphs. II. Single-exponential algorithms

TL;DR

This paper develops single-exponential algorithms for the -M-DELETION and -TM-DELETION problems on graphs with bounded treewidth, focusing on cases where  contains a single connected planar graph, advancing parameterized complexity understanding.

Contribution

It introduces single-exponential algorithms for specific -M-DELETION and -TM-DELETION problems on bounded treewidth graphs, expanding the class of problems with such algorithms.

Findings

Algorithms run in time 2^{O(tw)} nd n^{O(1)}.

Applicable to  containing certain small planar graphs.

Provides a basis for a complexity dichotomy in -M-DELETION.

Abstract

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION (resp. ${\cal F}$-TM-DELETION) problem consists in, given a graph $G$ and an integer $k$, decide 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 (resp. topological minor). We are interested in the parameterized complexity of both problems when the parameter is the treewidth of $G$, denoted by $tw$, and specifically in the cases where ${\cal F}$ contains a single connected planar graph $H$. We present algorithms running in time $2^{O(tw)} \cdot n^{O(1)}$, called single-exponential, when $H$ is either $P_3$, $P_4$, $C_4$, the paw, the chair, and the banner for both $\{H\}$-M-DELETION and $\{H\}$-TM-DELETION, and when $H=K_{1,i}$, with $i \geq 1$, for $\{H\}$-TM-DELETION. Some of these algorithms use the rank-based approach introduced by Bodlaender et al. [Inform Comput, 2015]. This is the second of a series of articles on this topic, and the results given here together with other ones allow us, in particular, to provide a tight dichotomy on the complexity of $\{H\}$-M-DELETION in terms of $H$.