Efficient dominating sets in labeled rooted oriented trees

In this paper, combinatorial enumeration techniques are used to derive the exact formulas for the number of efficient dominating sets among all labeled rooted oriented trees. These formulas are used to find the number of efficient dominating sets among all labeled rooted oriented trees of order p for each p up to 45. Finally, the asymptotic formulas for the number of efficient dominating sets among all labeled rooted oriented trees are also determined.