On the total k-domination number of graphs

Abstract

Let k be a positive integer and let G = (V, E) be a simple graph. The k-tuple domination number γ ×k (G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V , |N G [v] ∩ S| ≥ k. Also the total k-domination number γ ×k,t (G) of G is the minimum cardinality of a total k-dominating set S, a set that for every vertex v ∈ V , |N G (v) ∩ S| ≥ k. The k-transversal number τ k (H) of a hypergraph H is the minimum size of a subset S ⊆ V (H) such that |S ∩ e| ≥ k for every edge e ∈ E(H). We know that for any graph G of order n with minimum degree at least k, γ ×k (G) ≤ γ ×k,t (G) ≤ n. Obviously for every k-regular graph, the upper bound n is sharp. Here, we give a sufficient condition for γ ×k,t (G) < n. Then we characterize complete multipartite graphs G with γ ×k (G) = γ ×k,t (G). We also state that the total k-domination number of a graph is the k-transversal number of its open neighborhood hypergraph, and also the domination number of a graph is the transversal number of its closed neighborhood hypergraph. Finally, we give an upper bound for the total k-domination number of the cross product graph G × H of two graphs G and H in terms on the similar numbers of G and H. Also, we show that this upper bound is strict for some graphs, when k = 1.

New algorithms for weighted k-domination and total k-domination problems in proper interval graphs

“…The notion of k-domination was introduced by Fink and Jacobson in 1985 [25] and studied in a series of papers (e.g., [14,22,24,29,45]) and in a survey Chellali et al [13]. The notion of total k-domination was introduced by Kulli in 1991 [44] and studied under the name of k-tuple total domination by Henning and Kazemi in 2010 [34] and also in a series of recent papers [1,42,46,56]. The terminology "k-tuple total domination" was introduced in analogy with the notion of "k-tuple domination", introduced in 2000 by Harary and Haynes [31].…”

Improved Algorithms for k-Domination and Total k-Domination in Proper Interval Graphs

“…The notion of k-domination was introduced by Fink and Jacobson in 1985 [23] and studied in a series of papers (e.g., [14,20,22,27,42]) and in a survey by Chellali et al [13]. The notion of total k-domination was introduced by Kulli in 1991 [41] and studied under the name of k-tuple total domination by Henning and Kazemi in 2010 [32] and also in a series of recent papers [1,39,43,53]. The terminology "k-tuple total domination" was introduced in analogy with the notion of "k-tuple domination", introduced in 2000 by Harary and Haynes [29].…”

Subexponential fixed-parameter algorithms for partial vector domination

“…. , d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S ⊆ V such that every vertex v in V \ S (resp., in V ) has at least d(v) neighbors in S. These problems were introduced by [21], and they contain many existing problems, such as the dominating set, the total dominating set, the k-dominating set problem [16], and the total k-dominating set problem [28] (these k's are different from the solution size), and so on. Indeed, by setting d = (1, .…”