A theorem for an axiomatic approach to metric properties of graphs

Summary:If $G$ is a connected graph with distance function $d$, then by a step in $G$ is meant an ordered triple $(u, x, v)$ of vertices of $G$ such that $d(u, x) = 1$ and $d(u, v) = d(x, v) + 1$. A characterization of the set of all steps in a connected graph was published by the present author in 1997. In Section 1 of this paper, a new and shorter proof of that characterization is presented. A stronger result for a certain type of connected graphs is proved in Section 2