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 \(\ell \) 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.
Supported by EPSRC (EP/G043434/1), by a Scheme 7 grant from the London Mathematical Society, and by the German Research Foundation (KR 4286/1).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bonamy, M., Bousquet, N.: Recoloring bounded treewidth graphs. Electron. Notes Discrete Math. 44, 257–262 (2013)
Bonamy, M., Johnson, M., Lignos, I.M., Patel, V., Paulusma, D.: Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs. J. Comb. Optim. 27, 132–143 (2014)
Bonsma, P.: The Complexity of rerouting shortest paths. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 222–233. Springer, Heidelberg (2012)
Bonsma, P.: Rerouting shortest paths in planar graphs. In: D’Souza, D., Kavitha, T., Radhakrishnan, J. (eds.) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012), LIPIcs, pp. 337–349. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Wadern (2012)
Bonsma, P., Cereceda, L.: Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theor. Comput. Sci. 410, 5215–5226 (2009)
Bonsma, P., Mouawad, A.E.: The complexity of bounded length graph recolouring, Manuscript (2014). arXiv:1404.0337
Bonsma, P., Kamiński, M., Wrochna, M.: Reconfiguring independent sets in claw-free graphs. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 86–97. Springer, Heidelberg (2014)
Cereceda, L., van den Heuvel, J., Johnson, M.: Connectedness of the graph of vertex-colourings. Discrete Math. 308, 913–919 (2008)
Cereceda, L., van den Heuvel, J., Johnson, M.: Mixing 3-colourings in bipartite graphs. Eur. J. Comb. 30, 1593–1606 (2009)
Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between 3-colourings. J. Graph Theory 67, 69–82 (2010)
Gopalan, P., Kolaitis, P.G., Maneva, E.N., Papadimitriou, C.H.: The connectivity of boolean satisfiability: computational and structural dichotomies. SIAM J. Comput. 38, 2330–2355 (2009)
van den Heuvel, J.: The complexity of change. In: Blackburn, S.R., Gerke, S., Wildon, M. (eds.) Surveys in Combinatorics 2013. London Mathematical Society Lecture Note Series, pp. 127–160. Cambridge University Press, Cambridge (2013)
Ito, T., Kamiński, M., Demaine, E.D.: Reconfiguration of list edge-colorings in a graph. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 375–386. Springer, Heidelberg (2009)
Ito, T., Kawamura, K., Ono, H., Zhou, X.: Reconfiguration of list L(2,1)-labelings in a graph. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 34–43. Springer, Heidelberg (2012)
Ito, T., Kawamura, K., Zhou, X.: An improved sufficient condition for reconfiguration of list edge-colorings in a tree. In: Ogihara, M., Tarui, J. (eds.) TAMC 2011. LNCS, vol. 6648, pp. 94–105. Springer, Heidelberg (2011)
Ito, T., Demaine, E.D.: Approximability of the subset sum reconfiguration problem. In: Ogihara, M., Tarui, J. (eds.) TAMC 2011. LNCS, vol. 6648, pp. 58–69. Springer, Heidelberg (2011)
Kamiński, M., Medvedev, P., Milanič, M.: Complexity of independent set reconfigurability problems. Theor. Comput. Sci. 439, 9–15 (2012)
Kamiński, M., Medvedev, P., Milanič, M.: Shortest paths between shortest paths. Theor. Comput. Sci. 412, 5205–5210 (2011)
Mouawad A.E., Nishimura, N., Raman, V.: Vertex cover reconfiguration and beyond, Manuscript (2014) arXiv:1402.4926
Mouawad, A.E., Nishimura, N., Raman, V., Simjour, N., Suzuki, A.: On the parameterized complexity of reconfiguration problems. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 281–294. Springer, Heidelberg (2013)
Acknowledgements
We are grateful to several reviewers for insightful comments that greatly improved our presentation.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Johnson, M., Kratsch, D., Kratsch, S., Patel, V., Paulusma, D. (2014). Finding Shortest Paths Between Graph Colourings. In: Cygan, M., Heggernes, P. (eds) Parameterized and Exact Computation. IPEC 2014. Lecture Notes in Computer Science(), vol 8894. Springer, Cham. https://doi.org/10.1007/978-3-319-13524-3_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-13524-3_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-13523-6
Online ISBN: 978-3-319-13524-3
eBook Packages: Computer ScienceComputer Science (R0)