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 introduced by Nebeský (2005) in the context of step systems. Graphs with convex point-shadows and convex distance point-shadows also naturally occur in convexity theory. Brešar et al. (2026) recently showed that several classes of graphs are smooth and that smoothness is preserved by Cartesian products, gated amalgams, and isometric subgraphs. Weakly modular graphs comprise the most important classes of graphs from Metric Graph Theory: median, modular, Helly, bridged, and dual polar graphs. In this note, we characterize smooth and strongly smooth weakly modular graphs in terms of forbidden isometric subgraphs on 5 and 7 vertices. This settles Problem 1 of the paper by Brešar et al. We also characterize prime strongly smooth weakly modular graphs, i.e., strongly smooth weakly modular graphs that cannot be obtained from smaller graphs by Cartesian products and gated amalgams.

翻译：暂无翻译