  Semispaces of a convexity space $(X,C)$ are maximal convex sets missing a
point. The separation axiom $S_3$ asserts that any point $x_0\in X$ and any
convex set $A$ not containing $x_0$ can be separated by complementary
halfspaces (convex sets with convex complements) or, equivalently, that all
semispaces are halfspaces. In this paper, we study $S_3$ for geodesic convexity
in graphs and the structure of semispaces in $S_3$-graphs. We characterize
$S_3$-graphs and their semispaces in terms of separation by halfspaces of
vertices $x_0$ and special sets, called maximal $x_0$-proximal sets and in
terms of convexity of their mutual shadows $x_0/K$ and $K/x_0$. In $S_3$-graphs
$G$ satisfying the triangle condition (TC), maximal proximal sets are the
pre-maximal cliques of $G$ (i.e., cliques $K$ such that $K\cup\{ x_0\}$ are
maximal cliques). This allows to characterize the $S_3$-graphs satisfying (TC)
in a structural way and to enumerate their semispaces efficiently. In case of
meshed graphs (an important subclass of graphs satisfying (TC)), the
$S_3$-graphs have been characterized by excluding five forbidden subgraphs. On
the way of proving this result, we also establish some properties of meshed
graphs, which maybe of independent interest. In particular, we show that any
connected, locally-convex set of a meshed graph is convex. We also provide
several examples of $S_3$-graphs, including the basis graphs of matroids.
Finally, we consider the (NP-complete) halfspace separation problem, describe
two methods of its solution, and apply them to particular classes of graphs and
graph-convexities.
