Abstract
AbstractThe domatic number of a graph G, denoted by DN(G), is the maximum number k such that V can be partitioned into k disjoint dominating sets. The domatic partition problem is to find a partition of the vertices of G into DN(G) dominating sets. The k-domatic partition problem with fixed k is to find a partition of the vertices of G into k dominating sets. In this paper, we show that 3-domatic partition problem is NP-complete on planar bipartite graphs, and the domatic partition problem is NP-complete on co-bipartite graphs. We further show that the unique 3-domatic partition problem is NP-hard on general graphs. Moreover, we propose an O(n)-time algorithm on the 3-domatic partition problem for maximal planar graphs, and O(n 3)-time algorithms on the domatic partition problem for P 4-sparse graphs and tree-cographs, respectively.
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