Axiomatic characterization of the interval function of a graph

A fundamental notion in metric graph theory is that of the interval function I : V × V → 2V − {∅} of a (finite) connected graph G = (V,E), where I(u, v) = { w | d(u, w) + d(w, v) = d(u, v) } is the interval between u and v. An obvious question is whether I can be characterized in a nice way amongst all functions F : V × V → 2V − {∅}. This was done in [13, 14, 16] by axioms in terms of properties of the functions F . The authors of the present paper, in the conviction that characterizing the interval function belongs to the central questions of metric graph theory, return here to this result again. In this characterization the set of axioms consists of five simple, and obviously necessary, axioms, already presented in [9], plus two more complicated axioms. The question arises whether the last two axioms are really necessary in the form given or whether simpler axioms would do the trick. This question turns out