Complete description of forbidden subgraphs in the structural domination problem

A class D 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 G is D -dominated if there exists a dominating subgraph D ∈ D in G . A connected graph G is minimal non- D -dominated if it is not D -dominated but all of its proper connected induced subgraphs are. We will give a complete description for the minimal non- D -dominated graphs for a concise D . The proof uses two stronger results.