A characterization of block graphs

A block graph is a graph whose blocks are cliques. For each edge image of a graph image, let image denote the set of all vertices in image which are closer to image than image. In this paper we prove that a graph image is a block graph if and only if it satisfies two conditions: (a) The shortest path between any two vertices of image is unique; and (b) For each edge image, if image and image, then, and only then, the shortest path between image and image contains the edge image. 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].