Complete description of forbidden subgraphs in the structural domination problem

A class DD of graphs is concise   if it only contains connected graphs and is closed under taking connected induced subgraphs. This paper is concerned with concise classes of graphs. A graph GG is DD-dominated if there exists a dominating subgraph D∈DD∈D in GG. A connected graph GG is minimal non-DD-dominated if it is not DD-dominated but all of its proper connected induced subgraphs are. We will give a complete description for the minimal non-DD-dominated graphs for a concise DD. The proof uses two stronger results.