A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G-D$ has a neighbor in $D$, while $D$ is 2-dominating if every vertex in $V_G-D$ has at least two neighbors in $D$. A graph $G$ is a $D{D}_2$-graph if it has a pair $(D,D_2)$ of disjoint subsets of vertices such that $D$ is a dominating set, and $D_2$ is a 2-dominating set of $G$. Studies of first properties of the $D{D}_2$-graphs were initiated by Henning and Rall (2013). In this paper, we continue their study and complete their structural characterization of all $D{D}_2$-graphs. Next, we focus on minimal $D{D}_2$-graphs and provide the relevant characterization of that class of graphs as well. Additionally, we study optimization problems related to $D{D}_2$-graphs and non-$D{D}_2$-graphs, respectively. In particular, for a given $D{D}_2$-graph $G$, the purpose is to find a minimal spanning $D{D}_2$-graph of $G$ of minimum or maximum size. We show that both these problems are NP-hard. Finally, if $G$ is a graph which is not a $D{D}_2$-graph, we consider the question of how many edges must be added to $G$ or subdivided in $G$ to ensure the existence of a $D{D}_2$-pair in the resulting graph. The latter problem turned out to be polynomially tractable, while the former one is NP-hard.

一个子集 $d\subseteq V_G$ 是主要的 $G$ 如果每个顶点都在 $V_G-d$ 有一个邻居 $d$，而 $d$ 如果每个顶点都为2， $V_G-d$ 至少有两个邻居 $d$。图$G$ 是一个 $d{d}_2$-图，如果有一对 $（d，d_2）$ 顶点的不相交子集 $d$ 是一个主要的集合，并且 $d_2$ 是2的一组 $G$。研究第一个性质$d{d}_2$图由Henning和Rall（2013）发起。在本文中，我们将继续他们的研究并完成他们对所有物体的结构表征$d{d}_2$-图。接下来，我们关注最小$d{d}_2$-图，并提供该类图的相关特征。此外，我们研究了与$d{d}_2$-图和非$d{d}_2$图。特别是对于给定$d{d}_2$-图形 $G$，目的是找到最小的跨度 $d{d}_2$的图 $G$最小或最大尺寸。我们证明这两个问题都是NP难题。最后，如果$G$ 是不是 $d{d}_2$图，我们考虑必须添加多少条边的问题 $G$ 或细分为 $G$ 确保存在 $d{d}_2$对在结果图中。事实证明，后一个问题在多项式上可以解决，而前一个问题是NP难题。