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

In a graph $$G = (V,E)$$ G = ( V , E ) , a set $$S\subseteq V(G)$$ S ⊆ 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 $$v \in V(G)$$ v ∈ 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 $$w(S) = \sum _{v \in S} {w(v)}$$ w ( S ) = ∑ v ∈ S w ( v ) is minimized. In this paper, we propose an $$O(n+m)$$ 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.