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Lean Tree-Cut Decompositions: Obstructions and Algorithms

Authors Archontia C. Giannopoulou, O-joung Kwon, Jean-Florent Raymond , Dimitrios M. Thilikos

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Archontia C. Giannopoulou
  • LaS team, Technische Universität Berlin, Germany
O-joung Kwon
  • Department of Mathematics, Incheon National University, South Korea
Jean-Florent Raymond
  • LaS team, Technische Universität Berlin, Germany
Dimitrios M. Thilikos
  • AlGCo project-team, LIRMM, Université de Montpellier, CNRS, Montpellier, France

Funding

  • Giannopoulou, Archontia C.: Supported by the ERC consolidator grant Distruct-648527.
  • Kwon, O-joung: Supported by the ERC consolidator grant Distruct-648527 and by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education (No. NRF-2018R1D1A1B07050294).
  • Raymond, Jean-Florent: Supported by ERC consolidator grant Distruct-648527.
  • Thilikos, Dimitrios M.: Supported by projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010). Supported by the Research Council of Norway and the French Ministry of Europe and Foreign Affairs, via the Franco-Norwegian project PHC AURORA 2019.

Acknowledgements

We are grateful to Michał Pilipczuk and Marcin Wrochna for extensive discussions about the proof of Theorem 1.

Cite AsGet BibTex

Archontia C. Giannopoulou, O-joung Kwon, Jean-Florent Raymond, and Dimitrios M. Thilikos. Lean Tree-Cut Decompositions: Obstructions and Algorithms. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.STACS.2019.32

Abstract

The notion of tree-cut width has been introduced by Wollan in [The structure of graphs not admitting a fixed immersion, Journal of Combinatorial Theory, Series B, 110:47 - 66, 2015]. It is defined via tree-cut decompositions, which are tree-like decompositions that highlight small (edge) cuts in a graph. In that sense, tree-cut decompositions can be seen as an edge-version of tree-decompositions and have algorithmic applications on problems that remain intractable on graphs of bounded treewidth. In this paper, we prove that every graph admits an optimal tree-cut decomposition that satisfies a certain Menger-like condition similar to that of the lean tree decompositions of Thomas [A Menger-like property of tree-width: The finite case, Journal of Combinatorial Theory, Series B, 48(1):67 - 76, 1990]. This allows us to give, for every k in N, an upper-bound on the number immersion-minimal graphs of tree-cut width k. Our results imply the constructive existence of a linear FPT-algorithm for tree-cut width.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Fixed parameter tractability
Keywords
  • tree-cut width
  • lean decompositions
  • immersions
  • obstructions
  • parameterized algorithms

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