Hitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable

TL;DR

This paper proves that the problem of deleting vertices to eliminate certain topological minors in planar graphs is fixed parameter tractable, providing algorithms with exponential dependence on the parameter and polynomial dependence on the graph size.

Contribution

It establishes fixed parameter tractability for the extsc{${ m F}$-TM-Deletion} problem on planar graphs, with algorithms that extend to graphs on fixed surfaces.

Findings

Algorithms run in $2^{ ext{O}(k^2)} imes n^2$ time and $2^{ ext{O}(k)} imes n^4$ time.

The techniques extend to graphs embeddable on any fixed surface.

The problem is fixed parameter tractable on planar graphs.

Abstract

For a finite collection of graphs ${\cal F}$, the \textsc{${\cal F}$-TM-Deletion} problem has as input an $n$-vertex graph $G$ and an integer $k$ and asks whether there exists a set $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a topological minor. We prove that for every such ${\cal F}$, \textsc{${\cal F}$-TM-Deletion} is fixed parameter tractable on planar graphs. Our algorithm runs in a $2^{\mathcal{O}(k^2)}\cdot n^{2}$ time or, alternatively in $2^{\mathcal{O}(k)}\cdot n^{4}$ time. Our techniques can easily be extended to graphs that are embeddable on any fixed surface.