In this paper, we introduce the concept of k-power domination which is a common generalization of domination and power domination. We extend several known results for power domination to k-power domination. Concerning the complexity of the k-power domination problem, we first show that deciding whether a graph admits a k-power dominating set of size at most t is NP-complete for chordal graphs a...

In this paper we consider the (d, n)-domination number, γd,n(Qn), the distance-d domination number γd(Qn) and the connected distance-d domination number γc,d(Qn) of ndimensional hypercube graphs Qn. We show that for 2 ≤ d ≤ bn/2c, and n ≥ 4, γd,n(Qn) ≤ 2n−2d+2, improving the bound of Xie and Xu [19]. We also show that γd(Qn) ≤ 2n−2d+2−r, for 2 − 1 ≤ n − 2d + 1 < 2 − 1, and γc,d(Qn) ≤ 2n−d, for ...

The dual notions of domination and packing in finite simple graphs were first extensively explored by Meir and Moon in [15]. Most of the lower bounds for the domination number of a nontrivial Cartesian product involve the 2-packing, or closed neighborhood packing, number of the factors. In addition, the domination number of any graph is at least as large as its 2-packing number, and the invaria...

A caterpillar is a tree with the property that after deleting all its vertices of degree 1 a simple path is obtained. The signed 2-domination number γ s (G) and the signed total 2-domination number γ st(G) of a graph G are variants of the signed domination number γs(G) and the signed total domination number γst(G). Their values for caterpillars are studied.

A subset S ⊆ V in a graph G = (V,E) is a total [1, 2]-set if, for every vertex v ∈ V , 1 ≤ |N(v) ∩ S| ≤ 2. The minimum cardinality of a total [1, 2]-set of G is called the total [1, 2]-domination number, denoted by γt[1,2](G). We establish two sharp upper bounds on the total [1,2]-domination number of a graph G in terms of its order and minimum degree, and characterize the corresponding extrema...

Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number copies $H$ properly graph on $n$ vertices can contain it has no isomorphic to $F$. We determine order magnitude $\operatorname{ex}(n,C_s,$rainbow-$C_t)$ for $s,t$ with $s\not =3$. In particular, answer question Gerbner, M\...

We review a characterization of domination perfect graphs in terms of forbidden induced subgraphs obtained by Zverovich and Zverovich [12] using a computer code. Then we apply it to a problem of unique domination in graphs recently proposed by Fischermann and Volkmann. 1 Domination perfect graphs Let G be a graph. A set D ⊆ V (G) is a dominating set of G if each vertex of G either belongs to D ...

Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and ...

Rainbow vertex-connection number is the minimum k-coloring on vertex graph G and denoted by rvc(G). Besides, rainbow-vertex connection can be applied to some special graphs, such as prism path graph. Graph operation a method used create new combining two graphs. Therefore, this research uses corona product form at resulting from of (Pm,2 P3) (P3 Pm,2). The results study obtain that theorem rain...

A Roman domination function on a graph G = (V,E) is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman domination function f is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G, denoted by...