# The Complexity of Approximately Counting Retractions to Square-Free Graphs

## TL;DR

This paper classifies the computational complexity of approximately counting retractions and homomorphisms in square-free graphs, revealing a rich structure with some problems being as hard as counting independent sets.

## Contribution

It provides a complete complexity classification for approximately counting retractions to square-free graphs, including new IS-easiness results for homomorphisms.

## Key findings

- Identifies a class of graphs where counting retractions is IS-complete.
- Extends easiness results from retractions to homomorphisms.
- Settles the complexity of counting homomorphisms for several non-trivial graph classes.

## Abstract

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.