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