A Characterization of the Interval Function of a (Finite or Infinite) Connected Graph

The Forcing Convexity Number of a Graph

“…The closed intervals and convex sets in a connected graph were studied and characterized by Nebeský [6,7] and were also investigated extensively in the book by Mulder [5], where it was shown that these sets provide an important tool for studying metric properties of connected graphs. Convexity in graphs was also studied in [2,3,4].…”

On the distance function of a connected graph

“…Axiom (S) can be compared to Axiom VIII of the characterization of the set of all shortest paths in a connected graph in [3], to Axiom VIII of the characterization of the interval function of a finite connected graph in [4], to Axiom Y of the characterization of the interval function of an infinite connected graph in [6], and to Axiom G of the characterization of the set all steps in a finite connected graph in [5]. (By the interval function of a graph is meant the interval function in the sense of H. M. Mulder [2].…”

The All-Paths Transit Function of a Graph

“…The definition of I is in terms of the distance function of G. Nebeský [9], [10] has given an axiomatic characterization of the geodesic interval function without any reference to metric notions.…”

“…Motivated by the characterization of the geodesic interval function by Nebeský [9], [10] using axioms on I only, we present in this paper an axiomatic characterization of the all-paths function A. The all-paths function has a nice structure, reflecting the block cut-vertex structure of the graph.…”