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

Abstract

In a graph G = (V, E), a set S subset of 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 is an element of 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) = (Sigma nu is an element of 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.