Let $G = V, E$ be a simple connected undirected graph. A set $X \subseteq V$ is \emph{geodesically convex} if for any pair of vertices $x, y \in X$, all vertices on all shortest paths in $G$ from $x$ to $y$ are contained in $X$. A set $H \subseteq V$ is said to be a {halfspace} if both $H$ and its complement (denoted by $H^c$) are convex. Given two sets $A, B \subseteq V$, the { halfspace separation} problem asks if there exist complementary halfspaces $H, H^c$ such that $A \subseteq H$ and $B \subseteq H^c$. The halfspace separation problem is known to be NP-complete for the geodesic convexity of general graphs. We show that geodesic halfspace separation is polynomial for weakly bridged graphs, pseudo-modular graphs, and the basis graphs of matroids.

翻译：设 $G = V, E$ 是一个简单连通无向图。如果对于任意一对顶点 $x, y \in X$，图 $G$ 中从 $x$ 到 $y$ 的所有最短路径上的所有顶点都包含在 $X$ 中，则集合 $X \subseteq V$ 称为\emph{测地凸集}。集合 $H \subseteq V$ 被称为{半空间}，如果 $H$ 及其补集（记为 $H^c$）都是凸集。给定两个集合 $A, B \subseteq V$，{半空间分离}问题询问是否存在互补的半空间 $H, H^c$，使得 $A \subseteq H$ 且 $B \subseteq H^c$。已知对于一般图的测地凸性，半空间分离问题是 NP 完全的。我们证明，对于弱桥接图、伪模图以及拟阵的基图，测地半空间分离问题是多项式时间可解的。