Smooth Graphs

Abstract: The notion of smoothness was introduced originally in the context of step systems on connected graphs. Smoothness turns out to be a very general property of metrics defined by a five-point condition. Restricted to graphs, it is closely related to the convexity of point-shadows. We show that smoothness is preserved by isometric subgraphs, both Cartesian and strong graph products, and gated amalgams. As a consequence, median graphs and many of their generalizations are smooth. We also show that l1-graphs are smooth. On the other hand, an induced K2,3 or K1,1,3 is incompatible with smoothness. Finally, we characterize smooth graphs among the Ptolemaic graphs as precisely the K1,1,3-free Ptolemaic graphs.