SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 328-350, January 2020.
For a fixed graph H, we consider the H-Recoloring problem: given a graph G and two H-colorings of G, i.e., homomorphisms from G to H, can one be transformed into the other by changing one color at a time, maintaining an H-coloring throughout. This is the same as finding a path in the Hom(G,H) complex. For H=K_k this is the problem of finding paths between k-colorings, which was recently shown to be in P for kłeq 3 and PSPACE-complete otherwise. We generalize the positive side of this dichotomy by providing an algorithm that solves the problem in polynomial time for any H with no C_4 subgraph. This gives a large class of constraints for which finding solutions to the Constraint Satisfaction Problem is NP-complete but finding paths in the solution space is in P. The algorithm uses a characterization of possible reconfiguration sequences (paths in Hom(G,H)), whose main part is a purely topological condition described in terms of the fundamental groupoid of H seen as a topological space.

中文翻译：

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通过同态的同态重构

SIAM离散数学杂志，第34卷，第1期，第328-350页，2020年1月。
对于固定图H，我们考虑了H重着色问题：给定图G和G的两种H颜色，即从G到H的同态，可以通过一次更改一种颜色来将一种变换为另一种，整个H色。这与在Hom（G，H）复杂图中找到路径相同。对于H = K_k，这是在k色之间寻找路径的问题，最近，对于kłeq3，这在P中显示为P，否则，为PSPACE-complete。我们通过提供一种算法来解决任何没有C_4子图的H的多项式时间问题，从而概括了这种二分法的积极方面。这给出了一大类约束，对于这些约束，找到约束满足问题的解是NP完全的，但是在解空间中找到路径是在P中。