On the total k-domination number of graphs

Adel P. Kazemi. "On the total k-domination number of graphs." Discussiones Mathematicae Graph Theory 32.3 (2012): 419-426. <http://eudml.org/doc/270955>.

@article{AdelP2012,
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 τₖ(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.},
author = {Adel P. Kazemi},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {total k-domination (k-tuple total domination) number; k-tuple domination number; k-transversal number; total -domination number; -tuple total domination number; -tuple domination number; -transversal number},
language = {eng},
number = {3},
pages = {419-426},
title = {On the total k-domination number of graphs},
url = {http://eudml.org/doc/270955},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Adel P. Kazemi
TI - On the total k-domination number of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 3
SP - 419
EP - 426
AB - 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 τₖ(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.
LA - eng
KW - total k-domination (k-tuple total domination) number; k-tuple domination number; k-transversal number; total -domination number; -tuple total domination number; -tuple domination number; -transversal number
UR - http://eudml.org/doc/270955
ER -