For a fixed graph H, the H-Recoloring problem asks whether for two given homomorphisms from a graph G to H, we can transform one into the other by changing the image of a single vertex of G in each step and maintaining a homomorphism from G to H throughout. We extend an algorithm of Wrochna for H-Recoloring where H is a square-free loopless undirected graph to the more general setting of directed graphs. We obtain a polynomial-time algorithm for H-Recoloring in this setting whenever H is a loopless digraph that does not contain a 4-cycle of algebraic girth zero and whenever H is a reflexive digraph that contains neither a 3-cycle of algebraic girth 1 nor a 4-cycle of algebraic girth zero.

翻译：对于固定图H, H- Recoloring 问题询问,对于从图G到图H的两个给定同质体,我们是否可以通过将每步一个G顶点的图像从G改变成另一个G,并保持从G到H的同质性。我们将H- Recolor化的Wrochna算法(H是无方无环无环无方向的图形)扩展至更一般的定向图设置。当H是无循环的分解,不包含4周期的升格Girth 0,而H是反射分解法,既不包含3周期的升格Girth 1,也不包含4周期的升格Girth 。