Weakly modular graphs are defined as the class of graphs that satisfy the \emph{triangle condition ($TC$)} and the \emph{quadrangle condition ($QC$)}. We study an interesting subclass of weakly modular graphs that satisfies a stronger version of the triangle condition, known as the \emph{triangle diamond condition ($TDC$)}. and term this subclass of weakly modular graphs as the \emph{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 $I_G$ 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 $V\times V$ 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.

翻译：弱模图定义为满足三角形条件(TC)和四边形条件(QC)的图类。我们研究了弱模图的一个有趣子类，该类满足更强形式的三角形条件——三角形菱形条件(TDC)，并将此子类称为菱形-弱模图。观察到该类包含桥接图类和弱桥接图类。连通图G（顶点集为V）的区间函数I_G是度量图论中的重要概念，也是传递函数的典型范例；这是一种定义在笛卡尔积V×V上、取值于V的幂集，并满足扩张性、对称性和幂等性公理的集合函数。本文推导出一个有趣公理(J0')，它源自Marlow Sholander于1952年提出的著名公理(J0)。证明公理(J0')是菱形-弱模图的特征公理。我们提出任意传递函数R上若干独立的一阶中间性公理，并证明：任意传递函数R成为某个菱形-弱模图的区间函数当且仅当R满足这些中间性公理。对于桥接图和弱桥接图的区间函数，我们得到了类似的刻画结果。