Reconfiguration of Digraph Homomorphisms

TL;DR

This paper extends algorithms for the H-Recoloring problem to directed graphs, providing polynomial-time solutions for certain classes of loopless and reflexive digraphs based on cycle constraints.

Contribution

It generalizes an existing algorithm for undirected graphs to directed graphs, identifying specific conditions under which polynomial-time solutions are possible.

Findings

Polynomial-time algorithm for loopless digraphs without 4-cycle of algebraic girth zero.

Polynomial-time algorithm for reflexive digraphs without 3-cycle of algebraic girth 1 or 4-cycle of algebraic girth zero.

Extension of Wrochna's algorithm to a broader class of directed graphs.

Abstract

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.