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.