Abstract |
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 $$gamma(pi_{b}C_{n})geq cases{frac{n}{2}, &if $n = 4k ,$cr leftlceilfrac{n+1}{2} ight ceil, &if $n = 4k+2$,} mbox{and } gamma(pi_{b}C_{n})leq leftlfloor frac{5n+2}{8} ight floor.$$ |