A Characterization of Certain Ptolemaic Graphs

Abstract

With every connected graph G there is associated a metric space M(G) whose points are the vertices of the graph with the distance between two vertices a and b defined as zero if a = b or as the length of any shortest arc joining a and b if a ≠ b. A metric space M is called a graph metric space if there exists a graph G such that M = M (G), i.e., if there exists a graph G whose vertex set can be put in one-to-one correspondence with the points of M in such a way that the distance between every two points of M is equal to the distance between the corresponding vertices of G.

On the distance function of a connected graph

“…The next theorem is a reformulation of a theorem of D. C. Kay and G. Chartrand [1]. We will present an explicit proof of it.…”

“…D. C. Kay and G. Chartrand [1] found a necessary and sufficient condition for a metric f on V to be the distance function of a connected graph G with V (G) = V . In their theorem, the following axioms were used:…”

Graphs and Cubes

“…Prove Theorem 2.9 (Kay and Chartrand, 1965;Nebeský, 2008) 2.11. Describe the relation Θ for the path P n , the grid P 3 P 2 , the complete bipartite graphs K 2,2 and K 2,3 , and the complete graph K 4 .…”

Geodesics and steps in a connected graph

“…According to (81), sis 3 ->T S3. Since sis 2 ->T ^3, Axiom E implies that s 3 = s 2 and therefore, Sis 2 ->s s 3 . Thus (9i) holds.…”

“…Let G be a connected graph, and let d denote the distance function of G. (Note that in [3] a characterization of the distance function of a connected graph was given.) Obviously, if (1) is a walk in G, then d(vo,v n ) ^ n. By a geodesic (or a shortest path) in G we mean such a walk (1) that d(v 0 ,v n ) = n. It is not difficult to see that every geodesic in G is a path.…”