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

Computational Complexity;Discrete Mathematics