The Algorithmic Complexity of k-Domatic Partition of Graphs

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 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 ( 1 k + o(1)) lnn, generalizing the (1 + o(1)) lnn 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.