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.

Consider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of G or a blue copy of H for some fixed graphs G and H. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the on-line Ramsey numberr(G, H). In this paper, we consider the case where G is a path P k . We prove thatr(P 3 , P ℓ+1 ) = ⌈5ℓ/4⌉ =r(P 3 , C ℓ ) for all ℓ ≥ 5, and determiner(P 4 , P ℓ+1 ) up to an additive constant for all ℓ ≥ 3. We also prove some general lower bounds for on-line Ramsey numbers of the form r(P k+1 , H).

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.

Let G = (V, E) be a graph. Set D ⊆ V (G) is a total outerconnected dominating set of G if D is a total dominating set in G and G[V (G) − D] is connected. The total outer-connected domination number of G, denoted by γ tc (G), is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then γ tc (T) ≥ 2n 3. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.

For a given connected graph G = (V , E), a set D tr ⊆ V (G) is a total restrained dominating set if it is dominating and both D tr and V (G) − D tr do not contain isolate vertices. The cardinality of the minimum total restrained dominating set in G is the total restrained domination number and is denoted by tr (G). In this paper we characterize the trees with equal total and total restrained dominating numbers and give a lower bound on the total restrained dominating number of a tree T in terms of its order and the number of leaves of T .