a roman dominating function (rdf) on a graph g = (v,e) is defined to be a function satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. a set s v is a restrained dominating set if every vertex not in s is adjacent to a vertex in s and to a vertex in . we define a restrained roman dominating function on a graph g = (v,e) to be ...

A subset S ⊆ V in a graph G = (V,E) is a [1, 2]-set if for every vertex v ∈ V \ S, 1 ≤ |N(v)∩ S| ≤ 2, that is, every vertex v ∈ V \ S is adjacent to at least one but not more than two vertices in S. In this paper we relate the concept of [1, 2]-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings and k-depende...

The k-tuple domination problem, for a fixed positive integer k, is to find a minimum size vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The case when k 2 is called 2-tuple domination problem or double domination problem. In this paper, the 2-tuple domination problem is studied on interval graphs from an algorithmic point of view, which takes ...

Let G be a graph of order n ≥ 2 and n1, n2, .., nk be integers such that 1 ≤ n1 ≤ n2 ≤ .. ≤ nk and n1 + n2 + .. + nk = n. Let for i = 1, .., k: Ai ⊆ Kni where Km is the set of all pairwise non-isomorphic graphs of order m, m = 1, 2, ... In this paper we study when for a domination related parameter μ (such as domination number, independent domination number and acyclic domination number) is ful...

A k-dominating set is a set D k V such that every vertex i 2 V nD k has at least k i neighbours in D k. The k-domination number k (G) of G is the cardinality of a smallest k-dominating set of G. For k 1 = ::: = k n = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found then the conception of k-domina...

A Roman dominating function of a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. The Roman domination subdivision number sdγR(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order t...

This article proposes that South Africa, as multi-lingual country, has unique potential and that this potential is not to be found in some or other essence of what it means to be African, but in the daily struggles, frustrations and possibilities of life in a fragmented and divided multilingual society. In this fragmented and ‘impossible’ society there are moments (maybe rare moments) of true u...