The Interval function, Ptolemaic, distance hereditary, bridged graphs and axiomatic characterizations

TL;DR

This paper characterizes classes of graphs like Ptolemaic, distance hereditary, and bridged graphs through axioms on their interval functions, providing a unified axiomatic framework for these graph classes.

Contribution

It offers new axiomatic characterizations of the interval functions for these graph classes using betweenness axioms and transit functions.

Findings

Characterized Ptolemaic graphs via interval function axioms

Extended axiomatic framework to distance hereditary and bridged graphs

Provided a unified axiomatic approach using transit functions

Abstract

In this paper we consider certain types of betweenness axioms on the interval function $I_G$ of a connected graph $G$. We characterize the class of graphs for which $I_G$ satisfy these axioms. The class of graphs that we characterize include the important class of Ptolemaic graphs and some proper superclasses of Ptolemaic graphs: the distance hereditary graphs and the bridged graphs. We also provide axiomatic characterizations of the interval function of these classes of graphs using an arbitrary function known as \emph{transit function}.