  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 $\gamma(G)$ of $G$ is the minimum cardinality of a dominating
set of $G$. The common independence number $\alpha_c(G)$ 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
$\alpha(G)$ of $G$, that is, $\gamma(G) \le i(G) \le \alpha_c(G) \le
\alpha(G)$. A graph $G$ is domination perfect if $\gamma(H) = i(H)$ for every
induced subgraph $H$ of $G$. We define a graph $G$ as common domination perfect
if $\gamma(H) = \alpha_c(H)$ for every induced subgraph $H$ of $G$. We provide
a characterization of common domination perfect graphs in terms of ten
forbidden induced subgraphs.
