Hitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable

Abstract

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