Abstract
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.
Keywords
- Sensor Network
- Bipartite Graph
- Polynomial Time Algorithm
- Algorithmic Complexity
- Domination Number
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, and the National Natural Science Foundation of China Grant 61033001, 61061130540, 61073174.
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Liang, H. (2012). The Algorithmic Complexity of k-Domatic Partition of Graphs. In: Baeten, J.C.M., Ball, T., de Boer, F.S. (eds) Theoretical Computer Science. TCS 2012. Lecture Notes in Computer Science, vol 7604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33475-7_17
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DOI: https://doi.org/10.1007/978-3-642-33475-7_17
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