Partitioning H-Free Graphs of Bounded Diameter

Authors Christoph Brause, Petr Golovach , Barnaby Martin, Daniël Paulusma , Siani Smith

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Christoph Brause
  • Technische Universität Bergakademie Freiberg, Germany
Petr Golovach
  • University of Bergen, Norway
Barnaby Martin
  • Department of Computer Science, Durham University, UK
Daniël Paulusma
  • Department of Computer Science, Durham University, UK
Siani Smith
  • Department of Computer Science, Durham University, UK

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Christoph Brause, Petr Golovach, Barnaby Martin, Daniël Paulusma, and Siani Smith. Partitioning H-Free Graphs of Bounded Diameter. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of H-free graphs, that is, graphs that do not contain some graph H as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph H contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study (MFCS 2019) on the k-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the H-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to H-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • vertex partitioning problem
  • H-free
  • diameter
  • complexity dichotomy


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