We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

For a fixed positive integer k, a k-tuple total dominating set of a graph G is a subset D ⊆ V (G) such that every vertex of G is adjacent to at least k vertices in D. The k-tuple total domination problem is to determine a minimum k-tuple total dominating set of G. This paper studies k-tuple total domination from an algorithmic point of view. In particular, we present a linear-time algorithm for...

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...

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 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...

In combinatorics, a latin square is $$n\times n$$ matrix filled with n different symbols, each occurring exactly once in row and column. Associated to square, we can define simple graph called graph. this article, compute lower upper bounds for the domination number k-tuple total numbers of such graphs. Moreover, describe formula 2-tuple number.

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...

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,...

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...