Graphs with small additive stretch number

The additive stretch number ${\displaystyle s_{add}\left(G\right)}$ 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 ${\displaystyle s_{add}\left(G\right)\le k}$ for some k ∈ N₀ = {0,1,2,...}. Furthermore, we derive characterizations of these classes for k = 1 and k = 2.