A graph is γ_g-perfect exactly when it can be built from one vertex by adding complete graph components and by adding a vertex adjacent to everything except a perfect set of cliques; such graphs are recognizable in polynomial time.
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Perfect graphs for domination games
A graph is γ_g-perfect exactly when it can be built from one vertex by adding complete graph components and by adding a vertex adjacent to everything except a perfect set of cliques; such graphs are recognizable in polynomial time.