A linear time algorithm for optimal k-hop dominating set of a tree

We give a linear time algorithm to compute an optimal (minimum) k-hop dominating set D of a tree T for k≥1. This extends the previous result for an optimal 1-dominating set for trees. We use a rooted form T¯ of T, with an arbitrary node selected as the root, to direct the search for nodes of D in a bottom–up fashion. The decision whether to include a node x in D or not is based on the subtree of T¯ at x. Optimal k-hop dominating sets have many important practical applications.