Graph theory with applications to algorithms and computer science
On n-domination, n-dependence and forbidden subgraphs
Graph theory with applications to algorithms and computer science
A Study on r-Configurations---A Resource Assignment Problem on Graphs
SIAM Journal on Discrete Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Approximating the Domatic Number
SIAM Journal on Computing
Complexity of domination-type problems in graphs
Nordic Journal of Computing
Energy conservation via domatic partitions
Proceedings of the 7th ACM international symposium on Mobile ad hoc networking and computing
On k-domination and minimum degree in graphs
Journal of Graph Theory
Efficient clusterhead rotation via domatic partition in self-organizing sensor networks
Wireless Communications & Mobile Computing
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Let G=(V,E) be a simple undirected graph, and k be a positive integer. A k-dominating set of G is a set of vertices S⊆V satisfying that every vertex in V∖S is adjacent to at least k vertices in S. A k-domatic partition of G is a partition of V into k-dominating sets. The k-domatic number of G is the maximum number of k-dominating sets contained in a k-domatic partition of G. In this paper we study the k-domatic number from both algorithmic complexity and graph theoretic points of view. We prove that it is $\mathcal{NP}$-complete to decide whether the k-domatic number of a bipartite graph is at least 3, and present a polynomial time algorithm that approximates the k-domatic number of a graph of order n within a factor of $(\frac{1}{k}+o(1))\ln n$, generalizing the (1+o(1))ln n approximation for the 1-domatic number given in [5]. In addition, we determine the exact values of the k-domatic number of some particular classes of graphs.