Linear-Time Algorithm for the Paired-Domination Problem on Weighted Block Graphs

Given a graph G = (V,E), the domination problem is to find a minimum size vertex subset S ⊆ V (G) such that every vertex not in S is adjacent to a vertex in S. A dominating set S of G is called a paired-dominating set if the induced subgraph G[S] contains a perfect matching. The paired-domination problem involves finding a paired-dominating set S of G such that the cardinality of S is minimized. Suppose that, for each 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 weight w(S) = ∑ {w(v) : v ∈ S} 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 method. Moreover, the algorithm can be completed in O(n) time if the block-cut-vertex structure of G is given.