Smooth weakly modular graphs

2026-09-24

AI Summary

This study addresses the open problem posed by Brešar et al. concerning the structural characterization of smooth and strongly smooth classes within weakly modular graphs. By integrating techniques from metric graph theory, convexity theory, and Cartesian product and gated amalgamation decompositions, this work provides an exact characterization of weakly modular graphs through the exclusion of specific isometric subgraphs. Specifically, it establishes the first forbidden subgraph characterizations based on five- and seven-vertex configurations, formulates a definitive criterion for the smoothness of weakly modular graphs, and completes the full classification of prime strongly smooth weakly modular graphs. Collectively, these contributions resolve the aforementioned open problem, offering a comprehensive structural understanding of smoothness properties in weakly modular graphs.

Abstract

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.

Problem

• smooth graphs
• weakly modular graphs
• forbidden isometric subgraphs
• prime graphs
• Metric Graph Theory

Innovation

• Smooth weakly modular graphs
• Forbidden isometric subgraphs
• Strongly smooth graphs
• Prime graphs
• Metric Graph Theory