On two double type domination invariants of trees

In this paper, we study on double domination ([Formula: see text]) and double edge-vertex domination ([Formula: see text]) of trees. We obtain that determining the double [Formula: see text]-domination number is NP-complete for bipartite graphs. We also obtain that [Formula: see text] for a tree [Formula: see text] and as an implementation, we present the family of trees achieving the equality [Formula: see text].