A subset $D\subseteq V$ of a graph $G=(V,E)$ is called a global total k-dominating set of G if D is a total k-dominating set of both G and the complement $\bar{G}$ of G. The Minimum Global Total k-Domination problem is to find a global total k-dominating set of minimum cardinality of the input graph G and Decide Global Total k-Domination problem is the decision version of Minimum Global Total k-Domination problem. The Decide Global Total k-Domination problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen the NP-completeness result of Decide Global Total k-Domination problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs, star-convex bipartite graphs and doubly chordal graphs. On the positive side, we give a polynomial time algorithm for the Minimum Global Total k-Domination problem for chordal bipartite graphs, which is a subclass of bipartite graphs. We propose a $2(1+\ln |V|)$-approximation algorithm for the Minimum Global Total k-Domination problem for any graph. We show that Minimum Global Total k-Domination problem cannot be approximated within $(1-ϵ)\ln |V|$ for any $ϵ>0$ unless $\mathsf{\text{P}}=\mathsf{\text{NP}}$ for any integer $k\ge 1$. We further show that for bipartite graphs, Minimum Global Total k-Domination problem cannot be approximated within $(\frac{1}{6}-ϵ)\ln |V|$ for any $ϵ>0$ unless $\mathsf{\text{P}}=\mathsf{\text{NP}}$ for any $k\ge 1$. Finally, we show that the Minimum Global Total k-Domination problem is APX-complete for bounded degree bipartite graphs for any fixed integer $k\ge 1$.

一个子集 $d\subseteq V$ 图的 $G=（V，Ë）$如果D是G和补码的总k占优集，则称为G的全局总k占优集$\stackrel{〜}{G}$的ģ。在最小全球总 ķ -Domination问题是要找到一个全球性的总ķ -dominating组输入图形的最小基数的摹和决定全球总 ķ -Domination问题的判定型最低全球总 ķ -Domination问题。对于一般图以及二部图，“决定全局总 k-控制”问题都是NP-完全问题。在本文中，我们加强了决定全局总k-支配的NP-完备性结果 通过证明该问题仍然是NP-完全消除二分图，星形-凸二分图和双弦和弦图的问题。在积极方面，我们为弦二部图的最小全局总 k控制问题给出了多项式时间算法，该问题是二部图的子类。我们建议$2（\mathrm{1个}+\ln |V|）$最小全局总 k的-近似算法-任何图的控制问题。我们表明最小全局总 k-支配问题不能在以下范围内近似$（\mathrm{1个}-ϵ）\ln |V|$ 对于任何 $ϵ>0$ 除非 $\mathsf{\text{P}}=\mathsf{\text{NP}}$ 对于任何整数 $ķ\ge \mathrm{1个}$。我们进一步表明，对于二部图，最小全局总 k-支配问题不能在以下范围内近似$（\frac{\mathrm{1个}}{6}-ϵ）\ln |V|$ 对于任何 $ϵ>0$ 除非 $\mathsf{\text{P}}=\mathsf{\text{NP}}$ 对于任何 $ķ\ge \mathrm{1个}$。最后，我们证明了最低的全球总 ķ -Domination问题是APX -完成了有限度的二分图的任何固定的整数$ķ\ge \mathrm{1个}$。