  For a graph $G$, a function $\psi$ is called a \emph{bar visibility
representation} of $G$ when for each vertex $v \in V(G)$, $\psi(v)$ is a
horizontal line segment (\emph{bar}) and $uv \in E(G)$ iff there is an
unobstructed, vertical, $\varepsilon$-wide line of sight between $\psi(u)$ and
$\psi(v)$. Graphs admitting such representations are well understood (via
simple characterizations) and recognizable in linear time. For a directed graph
$G$, a bar visibility representation $\psi$ of $G$, additionally, puts the bar
$\psi(u)$ strictly below the bar $\psi(v)$ for each directed edge $(u,v)$ of
$G$. We study a generalization of the recognition problem where a function
$\psi'$ defined on a subset $V'$ of $V(G)$ is given and the question is whether
there is a bar visibility representation $\psi$ of $G$ with $\psi(v) =
\psi'(v)$ for every $v \in V'$. We show that for undirected graphs this problem
together with closely related problems are \NP-complete, but for certain cases
involving directed graphs it is solvable in polynomial time.
