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Common domination perfect graphs

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex that does not belong to $S$ is adjacent to a vertex in $S$. The domination number $γ(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. The common independence number $α_c(G)$ of $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set of cardinality at least~$r$. The common independence number is squeezed between the independent domination number $i(G)$ and the independence number $α(G)$ of $G$, that is, $γ(G) \le i(G) \le α_c(G) \le α(G)$. A graph $G$ is domination perfect if $γ(H) = i(H)$ for every induced subgraph $H$ of $G$. We define a graph $G$ as common domination perfect if $γ(H) = α_c(H)$ for every induced subgraph $H$ of $G$. We provide a characterization of common domination perfect graphs in terms of ten forbidden induced subgraphs.