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.
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Nebeský, L. On Properties of a Graph that Depend on its Distance Function. Czechoslovak Mathematical Journal 54, 445–456 (2004). https://doi.org/10.1023/B:CMAJ.0000042383.98585.97
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DOI: https://doi.org/10.1023/B:CMAJ.0000042383.98585.97