Abstract
For a graph H, the H-recolouring problem \({\text {Recol}}(H)\) asks, for two given homomorphisms from a given graph G to H, if one can get between them by a sequence of homomorphisms of G to H in which consecutive homomorphisms differ on only one vertex. We show that, if G and H are reflexive and H is triangle-free, then this problem can be solved in polynomial time. This shows, at the same time, that the closely related H-reconfiguration problem \({\text {Recon}}(H)\) of deciding whether two given homomorphisms from a given graph G to H are in the same component of the Hom-graph \({\text {{Hom}}}(G,H)\), can be solved in polynomial time for triangle-free reflexive graphs H.
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Notes
In contrast, it is interesting to remark that the problems \({{\,\mathrm{Recol}\,}}(H)\) and \({{\,\mathrm{Recon}\,}}(H)\) are not so closely related in the digraph case. For given H they may have different sets of connected YES instances; though we do not have examples yet where they have different complexity [3].
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We thank anonymous referees for their detailed reading of the paper. They pointed out several errors and made suggestions on presentation that greatly improved the paper.
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The third author was supported by Korean NRF Basic Science Research Program (2018-R1D1A1A09083741) funded by the Korean government (MEST) and the Kyungpook National University Research Fund.
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Lee, J.b., Noel, J.A. & Siggers, M. Recolouring homomorphisms to triangle-free reflexive graphs. J Algebr Comb 57, 53–73 (2023). https://doi.org/10.1007/s10801-022-01161-y
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DOI: https://doi.org/10.1007/s10801-022-01161-y