Independent and double domination in trees

Abstract

In a graph G, a vertex dominates itself and its neighbors. A subset S of vertices of G is a double dominating set if every vertex in G is dominated at least twice by the vertices of S. The minimum cardinality of a double dominating set of G is the double domination number. We determine sharp lower and upper bounds on the double domination number of a nontrivial tree, and characterize the trees attaining the bounds. As a consequence of these upper bounds, we are able to improve known bounds on the independent domination number of a tree.