在圖論中\textbf{支配問題}是對一張圖，
對此圖的點或邊做選取或給值，
使得這張圖能達到問題的要求(例如每個點都有鄰近的點被選取)。
在本論文中我們考慮兩個支配問題的變型，
其一是 extbf{有號支配問題}(Signed domination (SD) problem)其二是 extbf{帶負號支配問題}(Minus domination (MD) problem)。
其中，
前者是用一函數將圖中所有點給予$\{-1,+1\}$ 之值，
後者是用一函數將圖中所有點給予$\{-1,0,+1\}$ 之值，
此兩個問題的要求皆為:
對圖中任意一點，
其所有鄰近點及本身之值的合為正數，
並找出一個函數對此圖所有值的合為最小。

有號或帶負號支配問題可應用在社交網路中。
對有號支配問題，
我們可以有下面這樣假設：
假設贊成的意見為正一，
不贊成的意為為負一，
且要使每個人周圍的意見合為贊成(合為正數)，
這即是一個簡單的有號支配問題。
對帶負號問題，
我們可以用類似前面的假設，
但多一個不確定的意見為零，
依然要使每個人周圍的意見合為贊成(合為正數)，
即是帶負號支配問題。
這兩個問題在近幾年來廣泛被研究，
在時間複雜度的計算中，
我們常常可以將較複雜的問題簡化至此兩個問題，
以求得其時間複雜度。

最後，
在本論文中提供了以下研究成果：
在有號支配問題上：
(1) 在一般圖上SD問題是$W$[2]-hard。
(2) 在每點最多六個邊的圖上SD問題是APX-hard。
(3) 分別給與每點最多四個邊、五個邊及六個邊SD問題的近似解演算法。
(4) 在二分圖上SD問題是NP-complete。
(5) 在每點最多三個邊且為平面二分圖SD問題是NP-complete。
(6) 給予每點最多三個邊的圖一個固定參數可追蹤的演算法。
(7) 在基底為每點最高四個邊的universal-apex圖是NP-complete。
(8) 在universal-apex圖上SD問題是$W$[2]-hardness。
(9) 在基底為每點最高六個邊的universal-apex圖上是APX-hardness。
(10) 給予在基底為每點最高四個邊的universal-apex圖上的延伸演算法。
在帶負號支配問題上：
(1) 在一般圖上是$W$[2]-hard。
(2) 在每點最多七個邊的圖上是APX-hard。
(3) 給與每點最多三個邊的圖一個固定參數可追蹤的演算法。

A {\em domination problem} in graph theorem is that
in given graph, we choose some vertices or edges,
or assign values to vertices such that
the graph can accord with the rule of the problem.
In this paper, we study the algorithms and hardness for {\em Signed domination (SD) problem} and {\em Minus domination (MD) problem}.
The singed domination problem is that we assign value $\{-1,+1\}$ to all vertices,
the minus domination problem is that we assign value $\{-1,0,+1\}$ to all vertices,
both of two problems ask that for all vertices,
the sum value of the vertex and its neighbors is positive,
and find the function which contribute the minimum value of total vertices.

SD problem and MD problem can apply on social network,
let agree comments be positive one,
and disagree comments be negative one,
for anyone,
the sum comments of its neighbor are agree(sum value is positive),
it is a simple example of signed domination problem.
These two problem are studied widely in resent years,
in complexity compute,
we usually reduce harder problem to these two problem,
to find its time complexity.

In this paper, we give results as follows:
On signed domination problem:
(1) $W$[2]-hardness on general graphs.
(2) APX-hardness on maximum degree six graphs.
(3) Approximation algorithm on small degree graphs.
(4) NP-completeness on split graphs.
(5) NP-completeness on subcubic planar bipartite graphs.
(6) FPT-algorithm on subcubic graphs.
(7) NP-completeness for signed domination on degree-at-most-four-base universal-apex graphs.
(8) W[2]-hardness for SD problem on universal-apex graphs.
(9) APX-hardness for SD problem on degree-at-most-six-base universal-apex graphs.
(10) Exact-algorithm for SD problem on degree-at-most-four-base universal-apex graphs.
On minus domination problem:
(1) $W$[2]-hardness on general connected graphs.
(2) APX-hardness on graphs of max. degree 7.
(3) Fixed-parameter algorithm on subcubic graphs.

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