Intervals and steps in a connected graph

AbstractLet G be a (finite) connected graph. Intervals and steps in G are objects that depend on the distance function d of G. If u,v∈V(G), then by the u–v interval in G we mean the set{x∈V(G);d(u,x)+d(x,v)=d(u,v)}.By the interval function of G we mean the mapping I of V(G)×V(G) into the power set of V(G) such that I(u,v) is the u–v interval of G. By a step in G we mean an ordered triple (u,v,w) where u,v,w∈V(G), d(u,v)=1 and d(v,w)=d(u,w)−1. A characterization of the interval function of G and a characterization of the set of all steps in G were published by this author in 1994 and 1997, respectively.This paper is a review of author's results on intervals and steps in a connected graph. Some small results or short proofs are new