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Finding shortest paths between graph colourings.

Johnson, M. and Kratsch, D. and Kratsch, S. and Patel, V. and Paulusma, D. (2016) 'Finding shortest paths between graph colourings.', Algorithmica., 75 (2). pp. 295-321.


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

Item Type:Article
Keywords:Graph colouring, Graph algorithms, Reconfigurations, Reconfiguration graphs, Fixed parameter tractability.
Full text:(AM) Accepted Manuscript
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Publisher statement:The final publication is available at Springer via
Date accepted:30 April 2015
Date deposited:29 May 2015
Date of first online publication:12 May 2015
Date first made open access:12 May 2016

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