Let \(\gamma (G)\) and \(\gamma _{t}(G)\) be the domination number and the total domination number of a graph G, respectively, and let \(\gamma _g(G)\) and \(\gamma _{tg}(G)\) be the game domination number and the game total domination number of G, respectively. 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. All minimally \(\gamma _g\)-imperfect triangle-free graphs are determined. It is also proved that \(\gamma _{tg}\)-perfect graphs are precisely \(\overline{2P_3}\)-free cographs.

令\（\ gamma（G）\）和\（\ gamma _ {t}（G）\）分别是图G的控制数和总控制数，令\（\ gamma _g（G） \）和\（\伽马_ {TG}（G）\）的游戏控制数和游戏全控制数是G ^分别。然后，如果每个G的诱导子图F都满足\（\ gamma _g（F）= \ gamma（F，则G是\（\ gamma _g \） -完美（resp。\（\ gamma _ {tg} \） - perfect））\）（分别为\（\ gamma _ {tg}（F）= \ gamma _t（F）\））。得出\（\ gamma _g \） -完美图的递归表征。表征产生了\（\ gamma _g \）完美图的多项式识别算法。证明每一个最小\（\ gamma _g \）-不完美图的控制数为2。确定了所有最小的\（\ gamma _g \）-不完美的无三角形图。还证明了\（\ gamma _ {tg} \） -完美图恰好是\（\ overline {2P_3} \） -无共形。