# The Complexity of Counting Surjective Homomorphisms and Compactions

## TL;DR

This paper characterizes the computational complexity of counting surjective homomorphisms and compactions from an input graph to a fixed graph, extending previous work on homomorphism decision and counting problems.

## Contribution

It provides the first complete complexity classifications for counting surjective homomorphisms and compactions, advancing understanding of these graph mapping problems.

## Key findings

- Complete complexity classification for counting surjective homomorphisms.
- Complete complexity classification for counting graph compactions.
- Identification of a dichotomy for approximate counting in connected cases.

## Abstract

A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the vertices of H and all of the non-loop edges of H. Hell and Nesetril gave a complete characterisation of the complexity of deciding whether there is a homomorphism from an input graph G to a fixed graph H. A complete characterisation is not known for surjective homomorphisms or for compactions, though there are many interesting results. Dyer and Greenhill gave a complete characterisation of the complexity of counting homomorphisms from an input graph G to a fixed graph H. In this paper, we give a complete characterisation of the complexity of counting surjective homomorphisms from an input graph G to a fixed graph H and we also give a complete characterisation of the complexity of counting compactions from an input graph G to a fixed graph H. In an addendum we use our characterisations to point out a dichotomy for the complexity of the respective approximate counting problems (in the connected case).