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.
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Notes
PSPACE-completeness appears to be the default complexity for intractable instances of this kind of problem; see [16].
A (polynomial) compression is a relaxed form of (polynomial) kernelization: The output may be with respect to any (possibly unparameterized) problem.
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We are grateful to several reviewers for insightful comments that greatly improved our presentation.
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This study was 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).
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Johnson, M., Kratsch, D., Kratsch, S. et al. Finding Shortest Paths Between Graph Colourings. Algorithmica 75, 295–321 (2016). https://doi.org/10.1007/s00453-015-0009-7
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DOI: https://doi.org/10.1007/s00453-015-0009-7