Abstract. For a fixed graph [Formula: see text], the [Formula: see text]-Recoloring problem asks whether, given two homomorphisms from a graph [Formula: see text] to [Formula: see text], one homomorphism can be transformed into the other by changing the image of a single vertex in each step and maintaining a homomorphism to [Formula: see text] throughout. The most general algorithmic result for [Formula: see text]-Recoloring so far was proposed by Wrochna in 2014, who introduced a topological approach to obtain a polynomial-time algorithm for any undirected loopless square-free graph [Formula: see text]. We show that the topological approach can be used to recover essentially all previous algorithmic results for [Formula: see text]-Recoloring and that it is applicable also in the more general setting of digraph homomorphisms. In particular, we show that [Formula: see text]-Recoloring admits a polynomial-time algorithm if (i) [Formula: see text] is a loopless digraph that does not contain a 4-cycle of algebraic girth 0 and (ii) [Formula: see text] is a reflexive digraph that contains no triangle of algebraic girth 1 and no 4-cycle of algebraic girth 0. In both cases, we obtain a polynomial-time algorithm for finding shortest transformations.

For a fixed graph H, the H-Recoloring problem asks whether, given two homomorphisms from a graph G to H, one homomorphism can be transformed into the other by changing the image of a single vertex in each step and maintaining a homomorphism to H throughout. The most general algorithmic result for H-Recoloring so far has been proposed by Wrochna in 2014, who introduced a topological approach to obtain a polynomial-time algorithm for any undirected loopless square-free graph H. We show that the topological approach can be used to recover essentially all previous algorithmic results for H-Recoloring and that it is applicable also in the more general setting of digraph homomorphisms. In particular, we show that H-Recoloring admits a polynomial-time algorithm i) if H is a loopless digraph that does not contain a 4-cycle of algebraic girth 0 and ii) if H is a reflexive digraph that contains no triangle of algebraic girth 1 and no 4-cycle of algebraic girth 0.