Dynamic Programming on Bipartite Tree Decompositions

Lars Jaffke, Laure Morelle, Ignasi Sau, Dimitrios M. Thilikos

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We revisit a graph width parameter that we dub bipartite treewidth, along with its associated graph decomposition that we call bipartite tree decomposition. Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one vertex from the bipartite part of any other bag, while the width of such decomposition measures how far the bags are from being bipartite. Adapted from a tree decomposition originally defined by Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by Tazari [Theor. Comput. Sci. 2012], bipartite treewidth appears to play a crucial role for solving problems related to odd-minors, which have recently attracted considerable attention. As a first step toward a theory for solving these problems efficiently, the main goal of this paper is to develop dynamic programming techniques to solve problems on graphs of small bipartite treewidth. For such graphs, we provide a number of para-NP-completeness results, FPT-algorithms, and XP-algorithms, as well as several open problems. In particular, we show that Kt-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are FPT parameterized by bipartite treewidth. We also provide the following complexity dichotomy when H is a 2-connected graph, for each of the H-Subgraph- Packing, H-Induced-Packing, H-Scattered-Packing, and H-Odd-Minor-Packing problems: if H is bipartite, then the problem is para-NP-complete parameterized by bipartite treewidth while, if H is non-bipartite, then the problem is solvable in XP-time. Beyond bipartite treewidth, we define 1-H-treewidth by replacing the bipartite graph class by any graph class H. Most of the technology developed here also works for this more general parameter.

Original languageEnglish (US)
Title of host publication18th International Symposium on Parameterized and Exact Computation, IPEC 2023
EditorsNeeldhara Misra, Magnus Wahlstrom
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959773058
StatePublished - Dec 2023
Event18th International Symposium on Parameterized and Exact Computation, IPEC 2023 - Amsterdam, Netherlands
Duration: Sep 6 2023Sep 8 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference18th International Symposium on Parameterized and Exact Computation, IPEC 2023


  • bipartite graphs
  • dynamic programming
  • independent set
  • maximum cut
  • odd cycle transversal
  • odd-minors
  • packing
  • tree decomposition
  • vertex cover

ASJC Scopus subject areas

  • Software


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