On an extension of distance-hereditary graphs

Given a simple and finite connected graph G, the distance d G ( u , v ) is the length of the shortest induced {u,v}-path linking the vertices u and v in G. Bandelt and Mulder [H.J. Bandelt and H.M. Mulder, Distance Hereditary Graphs, J. Combin, Series B 41 (1986) 182–208] have characterized the class of distance hereditary graphs where the distance is preserved in each connected subgraph. In this paper, we are interested with the class of k-distance hereditary graphs ( k ∈ N ∗ ) which consists in a parametric extension of the distance-heredity notion. We allow the distance in each connected induced subgraph to increase by at most ( k − 1 ) unities. We provide a characterization of k-distance hereditary graphs in terms of forbidden configurations for each k ∈ N ∗ .