The Complexity of Counting Surjective Homomorphisms and Compactions
Jacob Focke, Leslie Ann Goldberg, Stanislav Živný · SIAM Journal on Discrete Mathematics · 2019
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 nonloop edges of $H$. Hell and Nešetřil gave a complete characterization of the complexity of deciding whether there is a homomorphism from an input graph $G$ to a fixed graph $H$. A complete characterization is not known for surjective homomorphisms or for compactions, though there are many interesting results. Dyer and Greenhill gave a complete characterization of the complexity of counting homomorphisms from an input graph $G$ to a fixed graph $H$. In this paper, we give a complete characterization of the complexity of counting surjective homomorphisms from an input graph $G$ to a fixed graph $H$, and we also give a complete characterization of the complexity of counting compactions from an input graph $G$ to a fixed graph $H$. In an addendum we use our characterizations to point out a dichotomy for the complexity of the respective approximate counting problems (in the connected case).