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A Complexity Dichotomy for Hitting Small Planar Minors Parameterized by Treewidth

Authors Julien Baste, Ignasi Sau , Dimitrios M. Thilikos



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Author Details

Julien Baste
  • Sorbonne Université, Laboratoire d'Informatique de Paris 6, LIP6, Paris, France
Ignasi Sau
  • LIRMM, CNRS, Université de Montpellier, Montpellier, France
Dimitrios M. Thilikos
  • LIRMM, CNRS, Université de Montpellier, Montpellier, France, and, Department of Mathematics, National and Kapodistrian University of Athens, Greece

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Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. A Complexity Dichotomy for Hitting Small Planar Minors Parameterized by Treewidth. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 2:1-2:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.IPEC.2018.2

Abstract

For a fixed graph H, we are interested in the parameterized complexity of the following problem, called {H}-M-Deletion, parameterized by the treewidth tw of the input graph: given an n-vertex graph G and an integer k, decide whether there exists S subseteq V(G) with |S| <= k such that G setminus S does not contain H as a minor. In previous work [IPEC, 2017] we proved that if H is planar and connected, then the problem cannot be solved in time 2^{o(tw)} * n^{O(1)} under the ETH, and can be solved in time 2^{O(tw * log tw)} * n^{O(1)}. In this article we manage to classify the optimal asymptotic complexity of {H}-M-Deletion when H is a connected planar graph on at most 5 vertices. Out of the 29 possibilities (discarding the trivial case H = K_1), we prove that 9 of them are solvable in time 2^{Theta (tw)} * n^{O(1)}, and that the other 20 ones are solvable in time 2^{Theta (tw * log tw)} * n^{O(1)}. Namely, we prove that K_4 and the diamond are the only graphs on at most 4 vertices for which the problem is solvable in time 2^{Theta (tw * log tw)} * n^{O(1)}, and that the chair and the banner are the only graphs on 5 vertices for which the problem is solvable in time 2^{Theta (tw)} * n^{O(1)}. For the version of the problem where H is forbidden as a topological minor, the case H = K_{1,4} can be solved in time 2^{Theta (tw)} * n^{O(1)}. This exhibits, to the best of our knowledge, the first difference between the computational complexity of both problems.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • parameterized complexity
  • graph minors
  • treewidth
  • hitting minors
  • topological minors
  • dynamic programming
  • Exponential Time Hypothesis

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References

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