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.

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