Domination using induced cliques in graphs (Q2760993)

A subset \(S\) of the vertex set \(V\) of a graph \(G\) is called \(H\)-independent for a certain class \(H\) of graphs, if the subgraph \(\langle S\rangle\) of \(G\) induced by \(S\) contains no subgraph isomorphic to a graph from \(H\). The set \(S\) is \(H\)-dominating, if for each \(v\in V-S\) the subgraph \(\langle S\cup\{v\}\rangle\) of \(G\) induced by \(S\cup\{v\}\) contains a subgraph containing \(v\) and isomorphic to a graph from \(H\). The minimum cardinality of an \(H\)-dominating (or \(H\)-independent and \(H\)-dominating simultaneously) set in \(G\) is the \(H\)-domination number \(\gamma_H(G)\) (or the \(H\)-independent domination number \(i_H(G)\) respectively). First the number \(\gamma_H(G)\) is studied in the general case. Then \(H\) equal to the class of all complete graphs is considered and the mentioned numerical invariants are studied for this case.