A characterization of block graphs

Abstract

A block graph is a graph whose blocks are cliques. For each edge e = u v of a graph G, let Ne (u) denote the set of all vertices in G which are closer to u than v. In this paper we prove that a graph G is a block graph if and only if it satisfies two conditions: (a) The shortest path between any two vertices of G is unique; and (b) For each edge e = u v ∈ E (G), if x ∈ Ne (u) and y ∈ Ne (v), then, and only then, the shortest path between x and y contains the edge e. This confirms a conjecture of Dobrynin and Gutman [A.A. Dobrynin, I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math., Beograd. 56 (1994) 18-22]. © 2009 Elsevier B.V. All rights reserved.