支配問題在圖形演算法上是很有名的問題， k 多重支配問題 就是其延伸。這個問題要求圖上的一個點不只要被一個點給支配，而是要被 k 個點支配。

針對樹的最小權值 k 多重支配問題，，拙作給了一個線性時間的演算法。針對平面圖，也證明其屬於 NP 完備集，此外也指出該證明之技巧不只對平面圖有用，還有很多圖類也可以被應用。

In a graph $G(V,E)$, a vertex $v$ dominates a vertex $u$ if $u=v$ or there is an edge from $v$ to $u$. A dominating set of $G$ is a subset $D$ of $V$ such that every vertex in $V$ is dominated by at least one vertex in $D$. Domination problem, that is proposed by K{\"o}nig, and its variations have fruitful literature more than $300$ publications. One class of those interesting variations is the {\it multiple domination problems}, i.e., each vertex in $V$ requires to be dominated by more than one vertex in $D$. In this thesis, we study the $k$-tuple domination problem on several graph classes. We give a linear time algorithm of weighted $k$-tuple domination problem, and prove NP-completeness of $k$-tuple domination problem on some graph classes like planar graphs.

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