  For graphs $G,H$, a homomorphism from $G$ to $H$ is an edge-preserving
mapping from $V(G)$ to $V(H)$. In the list homomorphism problem, denoted by
\textsc{LHom}($H$), we are given a graph $G$ and lists $L: V(G) \to 2^{V(H)}$,
and we ask for a homomorphism from $G$ to $H$ which additionally respects the
lists $L$.
  Very recently Okrasa, Piecyk, and Rz\k{a}\.zewski [ESA 2020] defined an
invariant $i^*(H)$ and proved that under the SETH $\mathcal{O}^*\left
(i^*(H)^{\textrm{tw}(G)}\right)$ is the tight complexity bound for
\textsc{LHom}($H$), parameterized by the treewidth $\textrm{tw}(G)$ of the
instance graph $G$. We study the complexity of the problem under dirretent
parameterizations. As the first result, we show that $i^*(H)$ is also the right
complexity base if the parameter is the size of a minimum feedback vertex set
of $G$.
  Then we turn our attention to a parameterization by the cutwidth
$\textrm{ctw}(G)$ of $G$. Jansen and Nederlof~[ESA 2018] showed that
\textsc{List $k$-Coloring} (i.e., \textsc{LHom}($K_k$)) can be solved in time
$\mathcal{O}^*\left (c^{\textrm{ctw}(G)}\right)$ where $c$ does not depend on
$k$. Jansen asked if this behavior extends to graph homomorphisms. As the main
result of the paper, we answer the question in the negative. We define a new
graph invariant $mim^*(H)$ and prove that \textsc{LHom}($H$) problem cannot be
solved in time $\mathcal{O}^*\left
((mim^*(H)-\varepsilon)^{\textrm{ctw}(G)}\right)$ for any $\varepsilon >0$,
unless the SETH fails. This implies that there is no $c$, such that for every
odd cycle the non-list version of the problem can be solved in time
$\mathcal{O}^*\left (c^{\textrm{ctw}(G)} \right)$.
  Finally, we generalize the algorithm of Jansen and Nederlof, so that it can
be used to solve \textsc{LHom}($H$) for every graph $H$; its complexity depends
on $\textrm{ctw}(G)$ and another invariant of $H$, which is constant for
cliques.
