The Number of Orientations of a Tree Admitting an Efficient Dominating Set

We give recursive methods for enumerating the number of orientations of a tree which can be efficiently dominated. We also examine the maximum number, \(\eta_q\), of orientations admitting an efficient dominating set in a tree with \(q\) edges. While we are unable to give either explicit formulas or recursive methods for finding \(\eta_q\), we are able to show that the growth rate of the sequence \(\langle\eta_q\rangle\) stabilizes by showing that \(\lim_{q\to\infty}\eta^\frac{1}{q}_q \) exists.