Graphs with small additive stretch number

The additive stretch number sadd(G) of a graph G is the maximum difference of the lengths of a longest induced path and a shortest induced path between two vertices of G that lie in the same component of G. We prove some properties of minimal forbidden configurations for the induced-hereditary classes of graphs G with sadd(G) ≤ k for some k ∈ N0 = {0, 1, 2, . . .}. Furthermore, we derive characterizations of these classes for k = 1 and k = 2.