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Then $G$ is $\\gamma_g$-perfect (resp. $\\gamma_{tg}$-perfect), if every induced subgraph $F$ of $G$ satisfies $\\gamma_g(F)=\\gamma(F)$ (resp. $\\gamma_{tg}(F)=\\gamma_t(F)$). A recursive characterization of $\\gamma_g$-perfect graphs is derived. The characterization yields a polynomial recognition algorithm for $\\gamma_g$-perfect graphs. It is proved that every minimally $\\gamma_g$-imperfect graph has domination number $2$. 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