A lower bound on the double outer-independent domination number of a tree

A vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph is a set of vertices of such that every vertex of is dominated by at least two vertices of , and the set ( ) \ is independent. The double outer-independent domination number of a graph , denoted by , is the minimum cardinality of a double outer-independent dominating set of . We prove that for every nontrivial tree of order , with leaves and support vertices we have , and we characterize the trees attaining this lower bound. We also give a constructive characterization of trees such that