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Note on "The Complexity of Counting Surjective Homomorphisms and Compactions"
Focke, Goldberg, and \v{Z}ivn\'y (arXiv 2017) prove a complexity dichotomy
for the problem of counting surjective homomorphisms from a large input graph G
without loops to a fixed graph H that may have loops. In this note, we give a
short proof of a weaker result: Namely, we only prove the #P-hardness of the
more general problem in which G may have loops. Our proof is an application of
a powerful framework of Lov\'asz (2012), and it is analogous to proofs of
Curticapean, Dell, and Marx (STOC 2017) who studied the "dual" problem in which
the pattern graph G is small and the host graph H is the input. Independently,
Chen (arXiv 2017) used Lov\'asz's framework to prove a complexity dichotomy for
counting surjective homomorphisms to fixed finite structures