An axiomatic approach to metric properties of connected graphs
Ladislav Nebeský · Czechoslovak Mathematical Journal · 2000
Let G be a nontrivial connected graph and let d denote its distance function.As is wellknown, d is a metric on V (G).In [4], and axiomatic characterization of the set of all geodesics (i.e.shortest paths) in G was given.In [8], an axiomatic characterization of the set of all steps in G (i.e. the set of all ordered triples (u, v, x) of vertices in G with the property that d(u, v) = 1 and d(v, x) = d(u, x) -1) was given.In the present paper, a certain connection between an axiomatic characterization of the set of all nontrivial geodesics in G and that of the set of all steps in G will be studied. 0.In this paper the letters i, j, k, m and n are reserved for denoting non-negative integers.By a graph we mean a finite undirected graph with no loop or multiple edge.In the whole paper we assume that a nontrivial connected graph G is given.Its vertex set, its edge set and its distance function will be denoted by V , E and d, respectively.Hence V is a finite set with at least two elements.As usual, if i 0, then V i+1 denotes the set of all ordered (i + 1)-tuples (1) (u 0 , . . ., u i ),where u 0 , . . ., u i ∈ V .Instead of (1) we will shortly write (2) u 0 . . .u i .If j 0, then we denote by Σ j the set