Weakly modular graphs with diamond condition, the interval function and axiomatic characterizations
<|reference_start|>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 \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.<|reference_end|>
@article{kamalolbhavan-sheela2024weakly,
    title={Weakly modular graphs with diamond condition, the interval function and
  axiomatic characterizations},
    author={Lekshmi Kamal Kamalolbhavan-Sheela, Jeny Jacob, Manoj Changat},
    journal={arXiv preprint arXiv:2403.01771},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.01771},
    primaryClass={math.CO cs.DM}
}