In a graph \(G = (V,E)\), a set \(S\subseteq V(G)\) is said to be a dominating set of G if every vertex not in S is adjacent to a vertex in S. Let G[S] denote the subgraph of G induced by a subset S of V(G). A dominating set S of G is called a paired-dominating set of G if the induced subgraph G[S] contains a perfect matching. Suppose that, for each \(v \in V(G)\), we have a weight w(v) specifying the cost for adding v to S. The weighted paired-domination problem is to find a paired-dominating set S whose total weights \(w(S) = \sum _{v \in S} {w(v)}\) is minimized. In this paper, we propose an \(O(n+m)\)-time algorithm for the weighted paired-domination problem on block graphs using dynamic programming, which strengthens the results in [Theoret Comput Sci 410(47–49):5063–5071, 2009] and [J Comb Optim 19(4):457–470, 2010]. Moreover, the algorithm can be completed in O(n) time if the block-cut-vertex structure of G is given.

在图\(G = (V,E)\) 中，如果每个不在S中的顶点都与S 中的一个顶点相邻，则称集合\(S\subseteq V(G)\)是G的支配集. 让G ^ [小号]分别表示的子图G ^由一个子集诱导的小号的V（G ^）。甲支配集小号的ģ被称为配对控制集的ģ如果导出子ģ [小号]包含一个完美匹配。假设，对于每个\(v \in V(G)\)，我们有一个权重w( v ) 指定将v添加到S的成本。加权成对支配问题是找到一个成对支配集S，其总权重\(w(S) = \sum _{v \in S} {w(v)}\)被最小化。在本文中，我们提出了一种\(O(n+m)\)时间算法，用于使用动态规划的块图上的加权配对支配问题，这加强了 [Theoret Comput Sci 410(47–49) 中的结果： 5063–5071, 2009] 和 [J Comb Optim 19(4):457–470, 2010]。此外，如果给定G的块切割顶点结构，该算法可以在O ( n )时间内完成。