A note on total domination (Q1065020)

A dominating [totally dominating] set is a subset D of the vertex set V(G) of a graph G with the property that for each \(x\in V(G)\setminus D\) [for each \(x\in V(G)]\) there exists \(y\in D\) adjacent to x. The domination number \(\gamma\) (G) [the total domination number \(\gamma_ t(G)]\) of G is the minimum number of vertices of a dominating [totally dominating] set in G. The independent domination number i(G) is the minimum number of vertices of a set in G which is simultaneously dominating and independent. An irredundant set in G is a set S with the property that the neighbourhood of any vertex of S is not a subset of the union of neighbourhoods of the other vertices of S. The minimum number of vertices of a maximal irredundant set in G is called the irredundance number ir(G) of G. The authors study interrelations among these numbers and further numerical invariants (independence number, vertex covering number, matching number, minimum matching number) of a graph.