Univerzita Karlova v Praze, Filozoficka fakulta, nam J. Palacha 2, 116 38 Praha 1, Czech Republic, e-mail: Ladislav.Nebesky@ff.cuni.cz
Abstract: 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.
Keywords: connected graphs, distance, steps, geodetically smooth graphs
Classification (MSC 2000): 05C12, 05C75
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