Finding topological subgraphs is fixed-parameter tractable

  • Authors:

  • Martin Grohe;Ken-ichi Kawarabayashi;Dániel Marx;Paul Wollan

  • Affiliations:

  • Humboldt-Universität zu Berlin, Berlin, Germany;National Institute of Informatics, Tokyo, Japan;Humboldt-Universität zu Berlin, Berlin, Germany;University of Rome, La Sapienza, Rome, Italy

  • Venue:
  • Proceedings of the forty-third annual ACM symposium on Theory of computing
  • Year:
  • 2011

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Abstract

We prove that for every fixed undirected graph H, there is an O(|V(G)|3) time algorithm that, given a graph G, tests 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.