Geodesic
On properties of a graph that depend on its distance function
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 2, pp. 445-456.

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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.
Classification : 05C12, 05C75
Keywords: connected graphs; distance; steps; geodetically smooth graphs 
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Nebeský, Ladislav. On properties of a graph that depend on its distance function. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 2, pp. 445-456. http://geodesic.mathdoc.fr/item/CMJ_2004__54_2_a16/