First-order logic axiomatization of metric graph theory

TL;DR

This paper develops a First-Order Logic with Betweenness axiomatization for various classes of graphs in Metric Graph Theory, highlighting both achievable and impossible cases.

Contribution

It provides a systematic FOLB axiomatization for many graph classes in Metric Graph Theory, extending Tarski's approach to this domain.

Findings

Axiomatization achieved for weakly modular and related graphs

Certain classes like chordal and planar graphs cannot be axiomatized in FOLB

Clarifies the logical foundations of metric graph classes

Abstract

The main goal of this note is to provide a First-Order Logic with Betweenness (FOLB) axiomatization of the main classes of graphs occurring in Metric Graph Theory, in analogy to Tarski's axiomatization of Euclidean geometry. We provide such an axiomatization for weakly modular graphs and their principal subclasses (median and modular graphs, bridged graphs, Helly graphs, dual polar graphs, etc), basis graphs of matroids and even $\Delta$-matroids, partial cubes and their subclasses (ample partial cubes, tope graphs of oriented matroids and complexes of oriented matroids, bipartite Pasch and Peano graphs, cellular and hypercellular partial cubes, almost-median graphs, netlike partial cubes), and Gromov hyperbolic graphs. On the other hand, we show that some classes of graphs (including chordal, planar, Eulerian, and dismantlable graphs), closely related with Metric Graph Theory, but defined in a combinatorial or topological way, do not allow such an axiomatization.