  Representing graphs by their homomorphism counts has led to the beautiful
theory of homomorphism indistinguishability in recent years. Moreover,
homomorphism counts have promising applications in database theory and machine
learning, where one would like to answer queries or classify graphs solely
based on the representation of a graph $G$ as a finite vector of homomorphism
counts from some fixed finite set of graphs to $G$. We study the computational
complexity of the arguably most fundamental computational problem associated to
these representations, the homomorphism reconstructability problem: given a
finite sequence of graphs and a corresponding vector of natural numbers, decide
whether there exists a graph $G$ that realises the given vector as the
homomorphism counts from the given graphs.
  We show that this problem yields a natural example of an
$\mathsf{NP}^{#\mathsf{P}}$-hard problem, which still can be $\mathsf{NP}$-hard
when restricted to a fixed number of input graphs of bounded treewidth and a
fixed input vector of natural numbers, or alternatively, when restricted to a
finite input set of graphs. We further show that, when restricted to a finite
input set of graphs and given an upper bound on the order of the graph $G$ as
additional input, the problem cannot be $\mathsf{NP}$-hard unless $\mathsf{P} =
\mathsf{NP}$. For this regime, we obtain partial positive results. We also
investigate the problem's parameterised complexity and provide fpt-algorithms
for the case that a single graph is given and that multiple graphs of the same
order with subgraph instead of homomorphism counts are given.
