A set $D \subseteq V$ of a graph $G=(V, E)$ is a dominating set of $G$ if each vertex $v\in V\setminus D$ is adjacent to at least one vertex in $D,$ whereas a set $D_2\subseteq V$ is a $2$-dominating (double dominating) set of $G$ if each vertex $v\in V \setminus D_2$ is adjacent to at least two vertices in $D_2.$ A graph $G$ is a $DD_2$-graph if there exists a pair ($D, D_2$) of dominating set and $2$-dominating set of $G$ which are disjoint. In this paper, we solve some open problems posed by M.Miotk, J.~Topp and P.{\.Z}yli{\'n}ski (Disjoint dominating and 2-dominating sets in graphs, Discrete Optimization, 35:100553, 2020) by giving approximation algorithms for the problem of determining a minimal spanning $DD_2$-graph of minimum size (Min-$DD_2$) with an approximation ratio of $3$; a minimal spanning $DD_2$-graph of maximum size (Max-$DD_2$) with an approximation ratio of $3$; and for the problem of adding minimum number of edges to a graph $G$ to make it a $DD_2$-graph (Min-to-$DD_2$) with an $O(\log n)$ approximation ratio. Furthermore, we prove that Min-$DD_2$ and Max-$DD_2$ are APX-complete for graphs with maximum degree $4$. We also show that Min-$DD_2$ and Max-$DD_2$ are approximable within a factor of $1.8$ and $1.5$ respectively, for any $3$-regular graph. Finally, we show the inapproximability result of Max-Min-to-$DD_2$ for bipartite graphs, that this problem can not be approximated within $n^{\frac{1}{6}-\varepsilon}$ for any $\varepsilon >0,$ unless P=NP.

翻译：设图 $G=(V, E)$ 中集合 $D \subseteq V$ 为 $G$ 的一个支配集，如果每个顶点 $v\in V\setminus D$ 至少与 $D$ 中一个顶点相邻；而集合 $D_2\subseteq V$ 为 $G$ 的一个 $2$-支配集（双支配集），如果每个顶点 $v\in V \setminus D_2$ 至少与 $D_2$ 中两个顶点相邻。若存在一对不相交的支配集与 $2$-支配集 $(D, D_2)$，则称图 $G$ 为 $DD_2$-图。本文针对 M.Miotk、J.~Topp 与 P.{\.Z}yl{i\'n}ski 在《Disjoint dominating and 2-dominating sets in graphs, Discrete Optimization, 35:100553, 2020》中提出的若干开放问题：确定最小尺寸的最小生成 $DD_2$-图（Min-$DD_2$）的近似比为 $3$；最大尺寸的最小生成 $DD_2$-图（Max-$DD_2$）的近似比为 $3$；以及向图 $G$ 添加最少边数使其成为 $DD_2$-图（Min-to-$DD_2$）的近似比为 $O(\log n)$，我们通过给出近似算法予以解决。此外，我们证明对于最大度为 $4$ 的图，Min-$DD_2$ 与 Max-$DD_2$ 是 APX-完全的。同时表明，任意 $3$-正则图的 Min-$DD_2$ 与 Max-$DD_2$ 可分别达到 $1.8$ 与 $1.5$ 的近似比。最后，我们给出二部图上 Max-Min-to-$DD_2$ 的不可近似性结果：除非 P=NP，否则对于任意 $\varepsilon >0$，该问题无法在 $n^{\frac{1}{6}-\varepsilon}$ 近似比内求解。