  In this paper we generalise the notion of visibility from a point in an
integer lattice to the setting of graph theory. For a vertex $x$ of a connected
graph $G$, we say that a set $S \subseteq V(G)$ is an \emph{$x$-position set}
if for any $y \in S$ the shortest $x,y$-paths in $G$ contain no point of
$S\setminus \{ y\}$. We investigate the largest and smallest orders of maximum
$x$-position sets in graphs, determining these numbers for common classes of
graphs and giving bounds in terms of the girth, vertex degrees, diameter and
radius. Finally we discuss the complexity of finding maximum vertex position
sets in graphs.
