Forbidden cycles in metrically homogeneous graphs

Abstract: In a paper by a superset of the authors it was proved that for every primitive 3-constrained space $\Gamma$ of finite diameter $\delta$ from Cherlin's catalogue of metrically homogeneous graphs, there exists a finite family $\mathcal F$ of ${1,\ldots, \delta}$-edge-labelled cycles such that a ${1,\ldots, \delta}$-edge-labelled graph is a subgraph of $\Gamma$ if and only if it contains no homomorphic images of cycles from $\mathcal F$. However, the cycles in the families $\mathcal F$ were not described explicitly as it was not necessary for the analysis of Ramsey expansions and the extension property for partial automorphisms. This paper fills this gap by providing an explicit description of the cycles in the families $\mathcal F$, heavily using the previous result in the process. Additionally, we explore the potential applications of this result, such as interpreting the graphs as semigroup-valued metric spaces or homogenizations of $\omega$-categorical ${1,\delta}$-edge-labelled graphs.