A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.

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弦图的一些子类中的半成对支配

一个没有孤立顶点的图$G$的支配集$D$被称为半成对支配集，如果$D$可以被划分为$2$-元素的子集，使得每个集合中的顶点的距离最多为$2$。半成对支配数，用$\gamma_{pr2}(G)$表示，是$G$半成对支配集的最小基数。给定一个没有孤立顶点的图 $G$，\textsc{Minimum Semipaired Domination} 问题是找到基数 $\gamma_{pr2}(G)$ 的 $G$ 的半成对支配集。\textsc{Minimum Semipaired Domination} 问题的决策版本对于弦图是一个重要的图类，已知它是 NP 完全的。在本文中，我们展示了 \textsc{Minimum Semipaired Domination} 问题的决策版本对于分裂图（和弦图的一个子类）仍然是 NP 完全的。从积极的方面来说，我们提出了一种线性时间算法来计算最小基数半成对支配块图集。此外，我们证明了 \textsc{Minimum Semipaired Domination} 问题对于最大度为 $3$ 的图是 APX 完全的。