The algorithmic complexity of k-domatic partition of graphs
TCS'12 Proceedings of the 7th IFIP TC 1/WG 202 international conference on Theoretical Computer Science
Randomized algorithms and upper bounds for multiple domination in graphs and networks
Discrete Applied Mathematics
On k-domination and j-independence in graphs
Discrete Applied Mathematics
Algorithmic aspects of the k-domination problem in graphs
Discrete Applied Mathematics
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A subset S of vertices of a graph G is k-dominating if every vertex not in S has at least k neighbors in S. The k-domination number $\gamma_k(G)$ is the minimum cardinality of a k-dominating set of G. Different upper bounds on $\gamma_{k}(G)$ are known in terms of the order n and the minimum degree $\delta$ of G. In this self-contained article, we present an Erdös-type result, from which some of these bounds follow. In particular, we improve the bound $\gamma_{k}(G) \le (2k- \delta - 1)n/(2k -\delta)$ for $(\delta +3)/2 \le k \le \delta - 1$, proved by Chen and Zhou in 1998. Furthermore, we characterize the extremal graphs in the inequality $\gamma_{k}(G) \le kn/(k +1)$, if $k \le \delta$, of Cockayne et al. This characterization generalizes that of graphs realizing $\gamma_1(G) = \gamma(G) = n/2$. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 33–40, 2008