Halfspace separation in geodesic convexity

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.