A note on an induced subgraph characterization of domination perfect graphs

Let \(\gamma(G)\) and \(\iota(G)\) be the domination and independent domination numbers of a graph \(G\), respectively. Introduced by Sumner and Moorer (1979), a graph \(G\) is domination perfect if \(\gamma(H) = \iota(H)\) for every induced subgraph \(H \subseteq G\). In 1991, Zverovich and Zverovich (1991)proposed a characterization of domination perfect graphs in terms of forbidden induced subgraphs. Fulman (1993) noticed that this characterization is not correct. Later, Zverovich and Zverovich (1995) 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 \(|Y| \leq |D|\).