Smooth Graphs
Bo\v{s}tjan Bre\v{s}ar, Manoj Changat, Prasanth G. Narasimha-Shenoi, Bruno J. Schmidt, Peter F. Stadler
TL;DR
This paper explores the concept of smoothness in graphs, showing its preservation under various graph operations and characterizing smooth graphs within certain classes.
Contribution
It introduces the notion of smoothness for graphs, proves its preservation under specific graph operations, and characterizes smooth graphs in Ptolemaic graphs.
Findings
Median graphs are smooth.
L1-graphs are smooth.
Certain subgraphs are incompatible with smoothness.
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.
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