The Algorithmic Complexity of k-Domatic Partition of Graphs

  • Hongyu Liang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7604)


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.


Sensor Network Bipartite Graph Polynomial Time Algorithm Algorithmic Complexity Domination Number 
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© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Hongyu Liang
    • 1
  1. 1.Institute for Interdisciplinary Information SciencesTsinghua UniversityBeijingChina

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