Geodesics and steps in a connected graph

On signpost systems and connected graphs

“…As was proved in [3] (and also in [5]), a simple signpost system is the step system of a connected graph if and only if (15) the underlying graph of S is connected and S satisfies the axiom (ALT) and a certain pair of additional axioms. Almost the same result was proved in [6]; but the proof in [6] is very different both from the proof in [3] and the proof in [5]; the pair of additional axioms was replaced by the following axiom (COM) in [6]:…”

“…Almost the same result was proved in [6]; but the proof in [6] is very different both from the proof in [3] and the proof in [5]; the pair of additional axioms was replaced by the following axiom (COM) in [6]:…”

Signpost systems and spanning trees of graphs

“…As was proved in [3], if S is a signpost system such that the underlying graph of S is connected, then S is the step system of a connected graph if and only if S satisfies a finite set A of certain axioms (all the axioms in A could be formulated in a language of the first order logic). A shorter proof of this result can be found in [6].…”

On Properties of a Graph that Depend on its Distance Function

“…Let G be a connected graph, and let d denote its distance function. Following [6], by a step in G we mean an ordered triple (u, x, v), where u, v, x ∈ V (G), d(u, x) = 1 and d(u, v) = d(x, v) + 1. If S denotes the set of all steps in G, then the ordered pair (V (G), S) is called the step system of G. The step system of G is a useful instrument for studying those properties of G that depend on its distance function.…”

“…Step systems were characterized by the present author in [6]. Using the notion of a signpost system, the characterization proved there can be described as follows:…”