  A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping
from $V(G)$ to $V(H)$. In the graph homomorphism problem, denoted by $Hom(H)$,
the graph $H$ is fixed and we need to determine if there exists a homomorphism
from an instance graph $G$ to $H$. We study the complexity of the problem
parameterized by the cutwidth of $G$.
  We aim, for each $H$, for algorithms for $Hom(H)$ running in time $c_H^k
n^{\mathcal{O}(1)}$ and matching lower bounds that exclude $c_H^{k \cdot
o(1)}n^{\mathcal{O}(1)}$ or $c_H^{k(1-\Omega(1))}n^{\mathcal{O}(1)}$ time
algorithms under the (Strong) Exponential Time Hypothesis.
  In the paper we introduce a new parameter that we call $\mathrm{mimsup}(H)$.
Our main contribution is strong evidence of a close connection between $c_H$
and $\mathrm{mimsup}(H)$:
  * an information-theoretic argument that the number of states needed in a
natural dynamic programming algorithm is at most $\mathrm{mimsup}(H)^k$,
  * lower bounds that show that for almost all graphs $H$ indeed we have $c_H
\geq \mathrm{mimsup}(H)$, assuming the (Strong) Exponential-Time Hypothesis,
and
  * an algorithm with running time $\exp ( {\mathcal{O}( \mathrm{mimsup}(H)
\cdot k \log k)}) n^{\mathcal{O}(1)}$.
  The parameter $\mathrm{mimsup}(H)$ can be thought of as the $p$-th root of
the maximum induced matching number in the graph obtained by multiplying $p$
copies of $H$ via certain graph product, where $p$ tends to infinity. It can
also be defined as an asymptotic rank parameter of the adjacency matrix of $H$.
Our results tightly link the parameterized complexity of a problem to such an
asymptotic rank parameter for the first time.
