Weakly modular graphs with diamond condition, the interval function and axiomatic characterizations
Lekshmi Kamal Kamalolbhavan-Sheela, Jeny Jacob, Manoj Changat
TL;DR
This work introduces diamond-weakly modular graphs, a subclass of weakly modular graphs defined by the triangle diamond condition $(TDC)$ alongside $(QC)$, and proves that they properly contain bridged and weakly bridged graphs while being closed under gated amalgamation. It provides an axiomatic characterization of the interval function via a weakened Sholander axiom $(J0')$, and situates this within the Mulder–Nebeský transit-function framework to connect arbitrary transit functions $R$ with the interval function $I_G$, yielding first-order definability results (FOLB) for these graph classes. The results establish a tight link between metric graph properties and interval/betweenness axioms, enabling precise identification of diamond-weakly modular, bridged, and weakly bridged graphs through axioms on $I_G$ (or on $R$). The paper also discusses structural closures (e.g., gated amalgamation) and non-closure under Cartesian products, situating diamond-weakly modular graphs within the broader landscape of metric graph theory and logical definability.
Abstract
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.