The total global domination number of a graph (Q1917733)

A subset \(D\) of the vertex set \(V(G)\) of a graph \(G\) is called dominating (or total dominating) in \(G\), if for each \(x \in V(G) - D\) (or for each \(x \in V(G)\) respectively) there exists \(y \in D\) adjacent to \(x\). A set \(D\) is called global dominating (or total global dominating) in \(G\), if it is dominating (or total dominating respectively) in both \(G\) and its complement \(\overline G\). The minimum number of vertices of a global dominating (or total global dominating) set in \(G\) is the global domination number \(\gamma_g (G)\) (or the total global domination number \(\gamma_{\text{tg}} (G)\) respectively). The total global domination number is studied. It is compared with other numerical invariants of graphs, namely with those related to the domination and with the vertex covering number. The graphs in which \(\gamma_{\text{tg}} (G)\) equals the number of vertices are characterized. The values of \(\gamma_{\text{tg}} (G)\) for paths, circuits and complete bipartite graphs are determined.