A note on an induced subgraph characterization of domination perfect graphs

Let γ(G) and ι(G) be the domination and independent domination numbers of a graph G, respectively. Introduced by Sumner and Moorer [23], a graph G is domination perfect if γ(H) = ι(H) for every induced subgraph H ⊆ G. In 1991, Zverovich and Zverovich [26] proposed a characterization of domination perfect graphs in terms of forbidden induced subgraphs. Fulman [15] noticed that this characterization is not correct. Later, Zverovich and Zverovich [27] offered such a second characterization with 17 forbidden induced subgraphs. However, the latter still needs to be adjusted. In this paper, we point out a counterexample. We then give a new characterization of domination perfect graphs in terms of only 8 forbidden induced subgraphs and a short proof thereof. Moreover, in the class of domination perfect graphs, we propose a polynomial-time algorithm computing, given a dominating set D, an independent dominating set Y such that |Y | ≤ |D|. keywords: domination, independent domination, forbidden induced subgraphs. MSC: 05C69, 05C75.