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, we prove that 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