假設 $G$
是一個圖，令$gamma(G)$是$G$的控制數，對於在$G$中任何點集合排列$pi$，
棱柱圖$pi C_{n}$是兩個同構於$G$的互斥圖，分別為$G_{1}$和$G_{2}$，
經由$pi(u)=v$連接$u$相對於$G_{1}$的點與$v$相對於$G_{2}$的點，所形成的圖。我們證明了，
對於所有的排列$pi$，
$$gamma(pi C_{n})geq cases{frac{ n}{ 2}, &若 $n = 4k ,$ cr
leftlceilfrac{n+1}{2} ight ceil, &若 $n
eq 4k$,} mbox{和 }
gamma(pi C_{n}) leq leftlceil frac{2n-1}{3} ight ceil mbox{。}$$
我們也找到一種排列$pi_{t}$，使得$gamma(pi_{t}C_{n})=k$，
其中$k$介於上述$gamma(pi C_{n})$的上下界之間。
最後，我們去證明，若$pi_{b}C_{n}$是一個二分部圖，那麼
$$gamma(pi_{b}C_{n})geq cases{frac{n}{2}, &若 $n = 4k ,$cr
leftlceilfrac{n+1}{2} ight ceil, &若 $n = 4k+2$,} mbox{和 }
gamma(pi_{b}C_{n})leq leftlfloor
frac{5n+2}{8} ight floor mbox{。}$$

Let $gamma(G)$ be the domination number of a graph $G$. For any
permutation $pi$ of the vertex set of a graph $G$, the prism of $G$
with respect to $pi$ is the graph $pi G$ obtained from two copies
$G_{1}$ and $G_{2}$ of $G$ by joining $uin V(G_{1})$ and $vin
V(G_{2})$ iff $v=pi(u)$. We prove that
$$gamma(pi C_{n})geq cases{frac{ n}{ 2}, &if $n = 4k ,$ cr
leftlceilfrac{n+1}{2} ight ceil, &if $n
eq 4k$,} mbox{and }
gamma(pi C_{n}) leq leftlceil frac{2n-1}{3} ight ceil
mbox{for all }pi.$$ We also find a permutation $pi_{t}$ such that
$gamma(pi_{t} C_{n})=k$, where $k$ between the lower bound and the
upper bound of $gamma(pi C_{n})$ in above. Finally, we prove that
if $pi_{b}C_{n}$ is a bipartite graph, then
$$gamma(pi_{b}C_{n})geq cases{frac{n}{2}, &if $n = 4k ,$cr
leftlceilfrac{n+1}{2} ight ceil, &if $n = 4k+2$,} mbox{and }
gamma(pi_{b}C_{n})leq leftlfloor frac{5n+2}{8} ight floor.$$

Contents
Abstract................................................................................1
1 Introduction......................................................................2
2 Previous results..............................................................4
3 The domination numbers of prisms of $C_{n}$.......5
References........................................................................17

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