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Algorithmica

, Volume 75, Issue 2, pp 295–321 | Cite as

Finding Shortest Paths Between Graph Colourings

  • Matthew Johnson
  • Dieter Kratsch
  • Stefan Kratsch
  • Viresh Patel
  • Daniël Paulusma
Article

Abstract

The \(k\)-colouring reconfiguration problem asks whether, for a given graph \(G\), two proper \(k\)-colourings \(\alpha \) and \(\beta \) of \(G\), and a positive integer \(\ell \), there exists a sequence of at most \(\ell +1\) proper \(k\)-colourings of \(G\) which starts with \(\alpha \) and ends with \(\beta \) and where successive colourings in the sequence differ on exactly one vertex of \(G\). We give a complete picture of the parameterized complexity of the \(k\)-colouring reconfiguration problem for each fixed \(k\) when parameterized by \(\ell \). First we show that the \(k\)-colouring reconfiguration problem is polynomial-time solvable for \(k=3\), settling an open problem of Cereceda, van den Heuvel and Johnson. Then, for all \(k \ge 4\), we show that the \(k\)-colouring reconfiguration problem, when parameterized by \(\ell \), is fixed-parameter tractable (addressing a question of Mouawad, Nishimura, Raman, Simjour and Suzuki) but that it has no polynomial kernel unless the polynomial hierarchy collapses.

Keywords

Graph colouring Graph algorithms Reconfigurations Reconfiguration graphs Fixed parameter tractability 

Notes

Acknowledgments

We are grateful to several reviewers for insightful comments that greatly improved our presentation.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Matthew Johnson
    • 1
  • Dieter Kratsch
    • 2
  • Stefan Kratsch
    • 3
  • Viresh Patel
    • 4
    • 5
  • Daniël Paulusma
    • 1
  1. 1.School of Engineering and Computing SciencesDurham University, Science LaboratoriesDurhamUK
  2. 2.Laboratoire d’Informatique Théorique et AppliquéeUniversité de LorraineMetz Cedex 01France
  3. 3.Institut für Softwaretechnik und Theoretische InformatikTechnische Universität BerlinBerlinGermany
  4. 4.School of Mathematical SciencesQueen Mary, University of LondonLondonUK
  5. 5.University of AmsterdamAmsterdamThe Netherlands

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