Let $kgeq 1$ be an integer, and let $G$ be a graph. A {it$k$-rainbow dominating function} (or a {it $k$-RDF}) of $G$ is afunction $f$ from the vertex set $V(G)$ to the family of all subsetsof ${1,2,ldots ,k}$ such that for every $vin V(G)$ with$f(v)=emptyset $, the condition $bigcup_{uinN_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ isthe open neighborhood of $v$. The {it weight} o...

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset in a graph such that every vertex in the graph is dominated by at least k vertices in this set. The current paper studies k-tuple domination in graphs from an algorithmic point of view. In particular, we give a linear...

We improve the generalized upper bound for the k-tuple domination number given in [A. Gagarin and V.E. Zverovich, A generalized upper bound for the k-tuple domination number, Discrete Math. 308 no. 5–6 (2008), 880–885]. Precisely, we show that for any graph G, when k = 3, or k = 4 and d ≤ 3.2, γ×k(G) ≤ ln(δ−k + 2) + ln ( (k − 2)d + ∑k−2 m=2 (k−m) 4min{m, k−2−m} d̂m + d̂k−1 ) + 1 δ − k + 2 n, and,...

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k , the k-tuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies the k-tuple domination problem in graphs from an algorithmic point of view. In particular, we give a line...

In this paper, a necessary and sufficient condition for the existence of an efficient 2-dominating set in a class of circulant graphs has been obtained and for those circulant graphs, an upper bound for the 2domination number is also obtained. For the circulant graphs Cir(n,A), where A = {1, 2, . . . , x, n − 1, n − 2, . . . , n − x} and x ≤ bn−1 2 c, the perfect 2-tuple total domination number...

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set, also known as a k-tuple total dominating set, is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, resp. total k-d...

Given a simple graph G = (V, E) and a fixed positive integer k. In a graph G, a vertex is said to dominate itself and all of its neighbors. A set D ⊆ V is called a k-tuple dominating set if every vertex in V is dominated by at least k vertices of D. The k-tuple domination problem is to find a minimum cardinality k-tuple dominating set. This problem is NP-complete for general graphs. In this pap...

a r t i c l e i n f o a b s t r a c t For a fixed positive integer k, a k-tuple total dominating set of a graph G = (V , E) is a subset T D k of V such that every vertex in V is adjacent to at least k vertices of T D k. In minimum k-tuple total dominating set problem (Min k-Tuple Total Dom Set), it is required to find a k-tuple total dominating set of minimum cardinality and Decide Min k-Tuple ...