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 #BIS. As retractions generalise homomorphisms, our easiness results extend to the important problem of approximately counting homomorphisms. By giving new #BIS-easiness results, we now settle the complexity of approximately counting homomorphisms for a whole class of non-trivial graphs that were previously unresolved.

中文翻译：

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对无平方图进行近似计数的复杂性

一种撤回是图的同态G到一个诱导子图H的G那是身份H. 在长期的研究中，已经在各种算法设置下研究了撤回。最近，考虑了近似计算撤回的问题。我们对所有无平方图（不包含长度为 4 的循环的图）的近似计算撤回的复杂性给出了完整的三分法。事实证明，有一个丰富而有趣的图类，这个问题在类#BIS 中是完整的。由于撤回概括了同态，我们的简单结果扩展到近似计算同态的重要问题。通过给出新的#BIS-easy 结果，我们现在解决了对先前未解决的整个非平凡图类近似计算同态的复杂性。