Common domination perfect graphs (Q6184333)

A subset \(S\) of vertices of a graph \(G\) is said to be an independent set if no two vertices in \(S\) are adjacent. A dominating set of a graph \(G\) is a set \(S\) of vertices of \(G\) such that every vertex of \(G\) is in \(S\) or has a neighbor in \(S\), where two vertices are neighbors in \(G\) if they are adjacent. An independent dominating set of a graph \(G\) is a dominating set of vertices that is also an independent set of \(G\). The independence number of a graph \(G\), denoted \(\alpha(G)\), is the cardinality of a maximum independent set of \(G\). The domination number of a graph \(G\), denoted \(\gamma(G)\), is the cardinality of a minimum dominating set of \(G\), while the independent domination number of \(G\), denoted \(i(G)\), is the cardinality of a minimum independent dominating set of \(G\). The common independence number of a graph \(G\), denoted by \(α_c(G)\), is the greatest integer \(r\) such that every vertex of \(G\) belongs to some independent set in \(G\) of cardinality at least \(r\). Motivated by the concept of perfect graphs in the chromatic sense, \textit{D. P. Sumner} and \textit{J. L. Moore} [``Domination perfect graphs'', Notices Am. Math. Soc. A-569 (1979)] defined a graph \(G\) to be domination perfect if \(\gamma(H)=i(H)\) for every induced subgraph \(H\) of \(G\). The authors define a graph \(G\) as common domination perfect if \(\gamma(H)=\alpha_c(H)\) for every induced subgraph \(H\) of \(G\). A characterization of common domination perfect graphs in terms of ten forbidden induced subgraphs is provided.