Let k ∈ Z +. A − distance Roman dominating function (kDRDF) on G = (V, E) is a f : V → {0, 1, 2} such that for every vertex v with f(v) 0, there u f(u) 2 d(u, v) ≤ k. The global (GkDRDF) if and only its complement G. weight of the value w(f) P x∈V f(x). minimum graph called domination number denoted as γ gR(G). gR(G) Note that, 1 usual γgR(G), is, γgR(G). authors initiated this study. In paper,...

Given a simple graph G = (V; E), a vertex v ∈ V is said to dominate itself and all vertices adjacent to it. A subset D of V is called an e cient dominating set of G if every vertex in V is dominated by exactly one vertex in D. The e cient domination problem is to 3nd an e cient dominating set of G with minimum cardinality. Suppose that each vertex v ∈ V is associated with a weight. Then, the we...

We study the parameterized complexity of domination-type problems. (σ, ρ)-domination is a general and unifying framework introduced by Telle: a set D of vertices of a graph G is (σ, ρ)dominating if for any v ∈ D, |N(v) ∩ D| ∈ σ and for any v / ∈ D, |N(v) ∩ D| ∈ ρ. We mainly show that for any σ and ρ the problem of (σ, ρ)-domination is W[2] when parameterized by the size of the dominating set. T...

The k-domination number of a graph is the minimum size of a set X such that every vertex of G is in distance at most k from X. We give a linear time constant-factor approximation algorithm for k-domination number in classes of graphs with bounded expansion, which include e.g. proper minor-closed graph classes, classes closed on topological minors or classes of graphs that can be drawn on a fixe...

The TAG adjunction operation operates by splitting a tree at one node, which we will call the adjunction site. In the resulting structure, the subtrees above and below the adjunction site are separated by, and connected with, the auxiliary tree used in the composition. As the adjunction site is thus split into two nodes, with a copy in each subtree, a natural way of formalizing the adjunction o...