  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.
