Star Graphs: Threaded Distance Trees and E-Sets

The distribution of distances in the star graph \( S_{T_n} \) (\(1 < n \in \mathbb{Z}\)) is established, and subsequently a threaded binary tree is obtained that realizes an orientation of \( S_{T_n} \) whose levels are given by the distances to the identity permutation, via a pruning algorithm followed by a threading algorithm. In the process, the distributions of distances of the efficient dominating sets of \( S_{T_n} \) are determined.