Hitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable

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.