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.