A linear-time algorithm for weighted paired-domination on block graphs

In a graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = (V,E)$$\end{document}, a set S⊆V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\subseteq V(G)$$\end{document} 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∈V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \in V(G)$$\end{document}, 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)=∑v∈Sw(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(S) = \sum _{v \in S} {w(v)}$$\end{document} is minimized. In this paper, we propose an O(n+m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n+m)$$\end{document}-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.