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)≤i(G)≤α_c(G)≤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.