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.

翻译：凸性空间 $(X,C)$ 的半空间是不包含某点的极大凸集。分离公理 $S_3$ 断言：任意点 $x_0\in X$ 与不包含 $x_0$ 的凸集 $A$ 均可被互补的半空间（即补集也为凸集的凸集）分离，等价地，所有半空间均为半空间。本文研究图论中测地凸性的 $S_3$ 性质及 $S_3$-图中半空间的结构。我们通过顶点 $x_0$ 与特殊集合（称为极大 $x_0$-近端集）的半空间分离性，以及互阴影集 $x_0/K$ 与 $K/x_0$ 的凸性，刻画了 $S_3$-图及其半空间。在满足三角形条件 (TC) 的 $S_3$-图 $G$ 中，极大近端集即为 $G$ 的预极大团（即满足 $K\cup\{ x_0\}$ 为极大团的团 $K$）。由此可结构性地刻画满足 (TC) 的 $S_3$-图，并高效枚举其半空间。对于网格图（满足 (TC) 的重要子类），$S_3$-图已通过排除五个禁用于图得到刻画。在证明该结果的过程中，我们还建立了网格图的一些性质，这些性质可能具有独立的研究价值。特别地，我们证明了网格图中任意连通局部凸集均为凸集。此外，我们给出了包括拟阵基图在内的若干 $S_3$-图实例。最后，我们考虑了（NP完全的）半空间分离问题，描述了两种求解方法，并将其应用于特定图类与图凸性中。