On Properties of a Graph that Depend on its Distance Function
Ladislav Nebeský · Czechoslovak Mathematical Journal · 2004
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.