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, 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 $f(H,k)\cdot n^{O(1)}$ time. More significantly, we show that on graphs excluding $H$ as a topological subgraph, Graph Isomorphism can be solved in time $n^{f(H)}$. 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.

💡 Research Summary

The paper by Grohe and Marx extends the celebrated Robertson‑Seymour structure theorem for H‑minor‑free graphs to the broader class of graphs that exclude a fixed graph H as a topological subgraph. The authors prove that for any fixed H, every graph G that does not contain H as a topological subgraph admits a tree‑decomposition (more precisely, a treelike decomposition) with the following property: each torso of the decomposition is either (i) “almost bounded‑degree”, meaning that all but a constant number of vertices have degree bounded by a constant depending only on H, or (ii) “almost embeddable” into a surface of bounded genus (again depending only on H), possibly after adding a bounded number of apex vertices and a bounded number of vortices. Moreover, such a decomposition can be computed in fixed‑parameter tractable time f(H)·|V(G)|^{O(1)}.

The authors then demonstrate how this structural insight yields algorithmic applications. First, they consider the Partial Dominating Set problem (given k, find k vertices whose closed neighbourhood is as large as possible). By performing bottom‑up dynamic programming on the tree‑decomposition, and using the fact that the problem can be solved in linear time on both bounded‑degree graphs and on almost‑embeddable graphs (the latter have bounded local tree‑width), they obtain an algorithm running in time f(k,H)·n^{O(1)}. This shows that Partial Dominating Set is fixed‑parameter tractable when parameterized by k on the class of H‑topological‑subgraph‑free graphs.

The second, more substantial application concerns Graph Isomorphism. For bounded‑degree graphs (Luks, Babai) and for H‑minor‑free graphs (Ponomarenko, later refined by Grohe and Marx), polynomial‑time isomorphism tests are known. However, a naïve bottom‑up approach fails for the broader class because a decomposition algorithm need not produce isomorphic decompositions for isomorphic inputs. To overcome this, the authors develop an “Invariant Decomposition Theorem”: they construct a treelike decomposition (a DAG‑based generalisation of a tree‑decomposition) that is canonical, i.e., isomorphic graphs receive isomorphic decompositions. They then prove a “Lifting Lemma” stating that if one has a canonical decomposition and a canonisation algorithm for each bag, one obtains a canonisation (and thus an isomorphism test) for the whole class. The bag‑level canonisation uses existing algorithms for bounded‑degree graphs and for H‑minor‑free graphs; the invariant decomposition supplies the necessary coordination.

The paper’s technical core consists of several components:

1. Local decomposition lemmas based on the theory of tangles, which identify highly connected regions and allow the graph to be split into the two desired types of parts.
2. Use of the Robertson‑Seymour structure theorem to further decompose the H‑minor‑free parts into almost‑embeddable pieces.
3. Algorithmic implementation of the decomposition, relying on known fixed‑parameter algorithms for H‑minor testing and for constructing the Robertson‑Seymour decomposition.
4. Definition of treelike decompositions as triples (T,σ,α) over a DAG, together with proofs that they can be computed invariantly.
5. The Lifting Lemma, which formalises the bottom‑up canonisation strategy for weighted relational structures, extending the approach beyond plain graphs.

The authors also discuss the limitations of the topological‑subgraph relation: unlike the minor relation, it is not a well‑quasi‑order, so there is no finite obstruction set. Nevertheless, by isolating the “almost bounded‑degree” component, they recover a clean structural description.

In summary, the paper provides a deep structural theorem for H‑topological‑subgraph‑free graphs, an FPT algorithm for Partial Dominating Set on this class, and a polynomial‑time isomorphism test (and canonisation) that unifies and generalises the known polynomial‑time cases for bounded‑degree and H‑minor‑free graphs. The work bridges graph minor theory, topological graph theory, and algorithmic graph isomorphism, opening avenues for further algorithmic exploitation of the new decomposition framework.