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On the Complexity of Computing Treebreadth

  • Guillaume Ducoffe
  • Sylvain Legay
  • Nicolas Nisse
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)

Abstract

During the last decade, metric properties of the bags of tree-decompositions of graphs have been studied. Roughly, the length and the breadth of a tree-decomposition are the maximum diameter and radius of its bags respectively. The treelength and the treebreadth of a graph are the minimum length and breadth of its tree-decompositions respectively. Pathlength and pathbreadth are defined similarly for path-decompositions. In this paper, we answer open questions of [Dragan and Köhler, Algorithmica 2014] and [Dragan, Köhler and Leitert, SWAT 2014] about the computational complexity of treebreadth, pathbreadth and pathlength. Namely, we prove that computing these graph invariants is NP-hard. We further investigate graphs with treebreadth one, i.e., graphs that admit a tree-decomposition where each bag has a dominating vertex. We show that it is NP-complete to decide whether a graph belongs to this class. We then prove some structural properties of such graphs which allows us to design polynomial-time algorithms to decide whether a bipartite graph, resp., a planar graph, has treebreadth one.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Guillaume Ducoffe
    • 1
    • 2
  • Sylvain Legay
    • 3
  • Nicolas Nisse
    • 1
    • 2
  1. 1.Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271Sophia AntipolisFrance
  2. 2.InriaSophia AntipolisFrance
  3. 3.LRI, Univ. Paris Sud, Université Paris-SaclayOrsayFrance

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