Smooth weakly modular graphs

A graph $G=(V,E)$ is called smooth (respectively, strongly smooth) if for any two vertices $u,v\in V$, the distance point-shadow $v|u := \{ x\in V: d(u,x)=d(u,v)+d(v,x)\}$, respectively, the point-shadow $v/u := \{ x\in V: v\in\mathrm{conv}(u,x)\}$, is geodesically convex. Smooth graphs have been in