Double total domination of graphs (Q1356756)

Let \(G\) be a graph with vertices \(V\) and edges \(E\). A set \(S\subset E\cup V\) is said to be a total dominating set of \(G\) if each vertex and edge of \(G\) is in \(S\), or adjacent or incident to a member of \(S\). The total domination number \(\alpha_2(G)\) is the cardinality of the smallest subset \(S\) which is a total dominating set of \(G\). Given \(S\) and \(T\), both subsets of \(E\cup V\), we say that \(T\) dominates \(S\) if each member of \(S\) is in \(T\), or adjacent or incident to a member of \(T\). Let \(\alpha_2(S)\) be the smallest cardinality of a set \(T\) dominating \(S\). Then the double total domination number \(\alpha_d(G)\) is the minimum value of \(\alpha_2(S)\) taken over all total dominating sets \(S\) of \(G\). Relations are proved between \(\alpha_2(G)\) and \(\alpha_d(G)\), and upper and lower bounds are given for \(\alpha_d(G)\).