The domination problem and its variants represent a classical domain within algorithmic graph theory. Among these variants, the paired-domination problem holds particular prominence due to its real-world implications in security and surveillance domains. Given an input graph $G$, the paired-domination problem involves identifying a minimum dominating set $D$ that induces a subgraph of $G$ with a perfect matching. Lin et al.~[\emph{Paired-domination problem on distance-hereditary graphs}, Algorithmica, 2020] previously presented a solution to this problem with a time complexity of $O(n^2)$. This paper significantly enhances their findings by introducing an $O(n+m)$-time algorithm. Furthermore, the time complexity of this algorithm can be reduced to $O(n)$ when provided with a decomposition tree for the graph $G$.

翻译：支配问题及其变体是算法图论中的一个经典领域。在这些变体中，配对支配问题因其在安防监控等领域的实际应用而尤为重要。给定输入图$G$，配对支配问题要求找到一个最小支配集$D$，使得$D$在$G$中诱导的子图包含完美匹配。Lin等人~[\emph{Paired-domination problem on distance-hereditary graphs}, Algorithmica, 2020]先前提出了一个时间复杂度为$O(n^2)$的解决方案。本文通过提出一个$O(n+m)$时间的算法，显著改进了他们的结果。此外，当给定图$G$的分解树时，该算法的时间复杂度可进一步降至$O(n)$。