Let k be a positive integer. A subset S of V (G) in a graph G is a k-tuple total dominating set of G if every vertex of G has at least k neighbors in S. The k-tuple total domination number γ×k,t(G) of G is the minimum cardinality of a k-tuple total dominating set of G. In this paper for a given graph G with minimum degree at least k, we find some sharp lower and upper bounds on the k-tuple tota...

The chromatic number χ(G) of a graph G is the minimum number of colours required to colour the vertices of G in such a way that no two adjacent vertices of G receive the same colour. A partition of V into χ(G) independent sets (called colour classes) is said to be a χpartition of G. A set S ⊆ V is called a dominating set of G if every vertex in V − S is adjacent to a vertex in S. A dominating s...

iii Acknowledgments I want to thank my thesis advisor Jörg Rothe for all of his support during the past four years. In the first place, I am deeply grateful to him for giving me the chance to be part of his research team. Without his great efforts, I would never have had the chance to work in the scientific community. Many inspiring and valuable discussions with him initiated fruitful ideas tha...

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {±1,±2, . . . ,±k} is said to be a signed star {k}-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. The signed star {k}-domination number of a graph G is γ{k}SS(G) = min{ ∑ e∈E f(e) | f is a S...

For a positive integer k, a k-rainbow dominating function of a digraph D is a function f from the vertex set V (D) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (D) with f(v) = ∅ the condition u∈N−(v) f(u) = {1, 2, . . . , k} is fulfilled, where N−(v) is the set of in-neighbors of v. A set {f1, f2, . . . , fd} of k-rainbow dominating functions on D with t...

A subset D of the vertex set V (G) of a graph G is called point-set dominating, if for each subset S ⊆ V (G) − D there exists a vertex v ∈ D such that the subgraph of G induced by S ∪ {v} is connected. The maximum number of classes of a partition of V (G), all of whose classes are point-set dominating sets, is the point-set domatic number dp(G) of G. Its basic properties are studied in the paper.