ON PROPERTIES OF A GRAPH THAT DEPEND ON ITS DISTANCE FUNCTION


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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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