Structure theorem and isomorphism test for graphs with excluded topological subgraphs

We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph H as a minor to graphs excluding H as a topological subgraph. We prove that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter |H|. We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a "typical" application of the structure theorem, we show that on graphs excluding H as a topological subgraph, Partial Dominating Set (find k vertices whose closed neighborhood has maximum size) can be solved in time f(H,k) • nO(1) time. More significantly, we show that on graphs excluding H as a topological subgraph, Graph Isomorphism can be solved in time nf(H). This result unifies and generalizes two previously known important polynomial-time solvable cases of Graph Isomorphism: bounded-degree graphs and H-minor free graphs. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition.