Structural domination of graphs

In a graph G = (V, E), a set S of vertices (as well as the subgraph induced by S) is said to be dominating if every vertex in V\S has at least one neighbor in S. For a given class D of connected graphs, it is an interesting problem to characterize the class Dom(D) of graphs G such that each connected induced subgraph of G contains a dominating subgraph belonging to V. Here we determine Dom(D) for D = (P-1, P-2, P-3), V = (K-t \ t greater than or equal to 1) boolean OR (P-3), and D = {connected graphs on at most four vertices} (where P-t and K-t denote the path and the complete graph on t vertices, respectively). The third theorem solves a problem raised by Cozzens and Kelleher [Discr. Math. 86 (1990), 101-116] It turns out that, in each case, a concise characterization in terms of forbidden induced subgraphs can be given.