Domination properties and induced subgraphs

Let the class Forb(Ct, Pt) consist of all graphs containing no induced cycle or path on t vertices, and denote by Dom(d, k) the class of graphs in which every connected induced subgraph H has a k-dominating subgraph D of diameter at most d (i.e. for each vertex x?V(H) ? V(D), there is a vertex y ? V(D) at distance ?k from x).In a previous paper we proved Forb(Ct, Pt) = Dom(t?4, 1) for 4?t?6 and Forb(Pt) = Dom(t?4, 1) for t?7. Here we show Forb(Ct, Pt?Dom(t?6, 2) for t?9; moreover, Forb(C9, P9) = Dom(3, 2) and Forb(C10, P10) = Dom(4, 2).