Weakly modular graphs with diamond condition, the interval function and axiomatic characterizations

Weakly modular graphs are defined as the class of graphs that satisfy the triangle condition ($TC$) and the quadrangle condition ($QC$). We study an interesting subclass of weakly modular graphs that satisfies a stronger version of the triangle condition, known as the triangle diamond condition ($TDC$). and term this subclass of weakly modular graphs as the diamond-weakly modular graphs. It is observed that this class contains the class of bridged graphs and the class of weakly bridged graphs. The interval function IG of a connected graph G with vertex set V is an important concept in metric graph theory and is one of the prime example of a transit function; a set function defined on the Cartesian product V× V to the power set of V satisfying the expansive, symmetric and idempotent axioms. In this paper, we derive an interesting axiom denoted as $(J0')$, obtained from a well-known axiom introduced by Marlow Sholander in 1952, denoted as $(J0)$. It is proved that the axiom $(J0')$ is a characterizing axiom of the diamond-weakly modular graphs. We propose certain types of independent first-order betweenness axioms on an arbitrary transit function R and prove that an arbitrary transit function becomes the interval function of a diamond-weakly modular graph if and only if R satisfies these betweenness axioms. Similar characterizations are obtained for the interval function of bridged graphs and weakly bridged graphs.