A set D of vertices of a graph G is a double dominating set of G if |N[v]∩D|≥2 for every v∈V(G), where N[v] represents the closed neighbourhood of v. The double domination number of G is the minimum cardinality among all double dominating sets of G. In this article, we show that if G and H are graphs with no isolated vertex, then for any vertex v∈V(H) there are six possible expressions, in terms of domination parameters of the factor graphs, for the double domination number of the rooted product graph G∘vH. Additionally, we characterize the graphs G and H that satisfy each of these expressions.

The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. We show that these formulas can be obtained relatively easily for the case of the first two parameters. The picture is quite different when it concerns the perfect Roman domination number. In this case, we obtain general bounds and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the lexicographic product graphs where the perfect Roman domination number equals the Roman domination number.

A subset D of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex of D. The total dominating set D is called a total co-independent dominating set if the subgraph induced by V−D is edgeless and has at least one vertex. The minimum cardinality of any total co-independent dominating set is the total co-independent domination number of G and is denoted by γt,coi(G)⁠. In this work we study some complexity and combinatorial properties of γt,coi(G)⁠. Specifically, we prove that deciding whether γt,coi(G)≤k for a given integer k is an NP-complete problem and give several bounds on γt,coi(G)⁠. Moreover, since any total co-independent dominating set is a total dominating set, we characterize all the trees having equal total co-independent domination number and total domination number.