近年來，在圖形理論中，支配數分割為一個經常深受探討的問題。在一個圖形上，將點分割成數個子集合，其中每個子集合皆為一個支配集，而我們將其最大的分割數稱之為支配數。支配數問題是在特定的圖形上找出它的支配數；支配數分割問題則實際在特定的圖形上找出最多組的支配集分割。若給定一個固定常數k，則k支配數分割問題是在特定的圖形上實際找出k組的支配集分割。另外，唯一支配數分割問題則是在特定圖形上判斷是否只有唯一一種支配集分割。
支配數分割問題已經被證明在一般圖形上為一個NP-complete複雜度的問題，在此篇論文中，我們證明在planar graphs及planar-bipartite graphs上的三支配數分割問題為一個NP-complete複雜度的問題，我們也證明在co-bipartite graphs上的支配數分割問題為一個NP-complete複雜度的問題，另外，我們還證明在一般圖形上的唯一支配數分割問題是一個NP-hard複雜度的問題。另一方面，我們提出幾個在特定圖形上解支配數分割的演算法，首先，我們提出一個演算法可在O(n) 時間內找出maximal planar graphs 上的支配數分割。接著，我們提出一個演算法可在O(n3) 時間內找出P4-sparse graphs上的支配數分割，最後，我們提出一個演算法可在O(n3) 時間內找出tree-cographs 上的支配數分割。

The domatic number of a graph G = (V,E), denoted by DN(G), is the maximum number k such that V can be partitioned into k disjoint dominating sets. The domatic number problem is to find DN(G) for a graph G. The domatic partition problem is to find a partition of the vertices of G into DN(G) disjoint dominating sets. The k-domatic partition problem with fixed k is to find a partition of the vertices of G into k disjoint dominating sets. The unique domatic partition problem is to decide whether G has an unique domatic partition or not. The unique k-domatic partition
problem with fixed k is to decide whether G has an unique k-domatic partition or not.
In this thesis, we show that 3-domatic partition problem is NP-complete on planar graphs and planar-bipartite graphs and the domatic partition problem is NP-complete on co-bipartite graphs. We also showed that the unique 3-domatic partition problem is NP-hard on general graphs. Moreover, we propose a 3-domatic partition algorithm for maximal planar graphs in O(n) time and O(n^3)-time algorithms for the domatic partition problem on P4-sparse graphs and tree-cographs, respectively.

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