Finding topological subgraphs is fixed-parameter tractable

Summary

We prove that for every fixed undirected graph H, there is an O(|V(G)|3) time algorithm that, given a graph G, tests if G contains H as a topological subgraph (that is, a subdivision of H is subgraph of G). This shows that topological subgraph testing is fixed-parameter tractable, resolving a longstanding open question of Downey and Fellows from 1992. As a corollary, for every H we obtain an O(|V(G)|3) time algorithm that tests if there is an immersion of H into a given graph G. This answers another open question raised by Downey and Fellows in 1992.