The Complexity of Approximately Counting Retractions to Square-free Graphs

A retraction is a homomorphism from graph G to an induced subgraph H of that the identity on . In long line research, retractions have been studied under various algorithmic settings. Recently, problem approximately counting was considered. We give complete trichotomy for complexity all square-free graphs (graphs do not contain cycle length 4). It turns out there rich and interesting class which this in #BIS. As generalise homomorphisms, our easiness results extend important homomorphisms. By giving new #BIS-easiness results, we now settle homomorphisms whole non-trivial were previously unresolved.