For a given connected graph G = (V, E), a set D ⊆ V (G) is a doubly connected dominating set if it is dominating and both D and V (G) − D are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.

The domination subdivision number sd(G) of a graph G is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of G. It has been shown [9] that sd(T ) ≤ 3 for any tree T . We prove that the decision problem of the domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the domination multisubdivision number of a nonempty graph G as a minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. We show that msd(G) ≤ 3 for any graph G. The domination subdivision number and the domination multisubdivision numer of a graph are incomparable in general case, but we show that for trees these two parameters are equal. We also determine domination multisubdivision number for some classes of graphs.

For a connected graph G = (V, E), a set D ⊆ V (G) is a dominating set of G if every vertex in V (G) − D has at least one neighbour in D. The distance d G (u, v) between two vertices u and v is the length of a shortest (u − v) path in G. An (u − v) path of length d G (u, v) is called an (u − v)-geodesic. A set X ⊆ V (G) is convex in G if vertices from all (a − b)-geodesics belong to X for any two vertices a, b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number γ con (G) of a graph G is the minimum cardinality of a convex dominating set in G. Graphs with the convex domination number close to their order are studied. The convex domination number of a Cartesian product of graphs is also considered.

In this paper, we study a generalization of the paired domination number. Let G = (V , E) be a graph without an isolated vertex. A set D ⊆ V (G) is a k-distance paired dominating set of G if D is a k-distance dominating set of G and the induced subgraph D has a perfect matching. The k-distance paired domination number k p (G) is the cardinality of a smallest k-distance paired dominating set of G. We investigate properties of the k-distance paired domination number of a graph. We also give an upper bound and a lower bound on the k-distance paired domination number of a non-trivial tree T in terms of the size of T and the number of leaves in T and we also characterize the extremal trees.

Let G = (V, E) be a graph with δ(G) ≥ 1. A set D ⊆ V is a paired dominating set if D is dominating, and the induced subgraph D contains a perfect matching. The paired domination number of G, denoted by γ p (G), is the minimum cardinality of a paired dominating set of G.