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Structure theorem and isomorphism test for graphs with excluded topological subgraphs
 In Proc. 44th ACM Symp. on the Theory of Computing
, 2012
"... 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 emb ..."
Abstract

Cited by 27 (2 self)
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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 fixedparameter 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 polynomialtime solvable cases of Graph Isomorphism: boundeddegree graphs [18] and Hminor free graphs [22]. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition.