Finding topological subgraphs is fixed-parameter tractable

We show that for every fixed undirected graph $H$, there is a $O(|V(G)|^3)$ time algorithm that tests, given a graph $G$, 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

Graphs avoiding immersion of K_{3,3}

DeVos and Malekian gave a structural description of graphs avoiding an immersion of K_{3,3}, showing that all such graphs are composed over small edge-cuts from graphs with at most 8 vertices and from 3-regular planar graphs. We provide another proof of this fact, simpler in some aspects.Comment: 18 pages, 1 figur