A Characterization of the Interval Function of a (Finite or Infinite) Connected Graph

By the interval function of a finite connected graph we mean the interval function in the sense of H. M. Mulder. This function is very important for studying properties of a finite connected graph which depend on the distance between vertices. The interval function of a finite connected graph was characterized by the present author. The interval function of an infinite connected graph can be defined similarly to that of a finite one. In the present paper we give a characterization of the interval function of each connected graph.