The interval function of a connected graph and road systems

Let V be a finite nonempty set. In this paper, a road system on V (as a generalization of the set of all geodesics in a connected graph G   with V(G)=VV(G)=V) and an intervaloid function on V (as a generalization of the interval function (in the sense of Mulder) of a connected graph G   with V(G)=VV(G)=V) are introduced. A natural bijection of the set of all intervaloid functions on V onto the set of all road systems on V is constructed. This bijection enables to deduce an axiomatic characterization of the interval function of a connected graph G from a characterization of the set of all geodesics in G.