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 $|Y| \leq |D|$.