Characterizations of Graphs with Stretch Number less than 2

Given a simple and finite connected graph G, the distance d G ( u , v ) is the length of the shortest ( u , v ) -path in G. Cicerone and Di Stefano [Graphs with bounded induced distance, Discrete Applied Mathematics 108 (2001), pp. 3–21] introduced and studied the class of k-distance-hereditary graphs, i.e., graphs where the distance in each connected induced subgraph is at most k times the distance in the whole graph. In this paper we make a step forward in the study of such graphs by providing characterizations for k-distance-hereditary graphs, k > 2 , in terms of both forbidden subgraphs and cycle-chord conditions. Such results lead to a polynomial-time recognition algorithm.