Finding topological subgraphs is fixed-parameter tractable

We prove that for every fixed undirected graph <i>H</i>, there is an O(|V(G)|³) time algorithm that, given a graph <i>G</i>, tests if <i>G</i> contains <i>H</i> as a topological subgraph (that is, a subdivision of <i>H</i> is subgraph of <i>G</i>). 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 <i>H</i> we obtain an O(|V(G)|³) time algorithm that tests if there is an immersion of <i>H</i> into a given graph <i>G</i>. This answers another open question raised by Downey and Fellows in 1992.