Star graphs: threaded distance trees and E-sets

TL;DR

This paper analyzes the distance distribution in star graphs, constructs a threaded binary tree representing these distances, and examines the properties of efficient dominating sets within these graphs.

Contribution

It introduces a novel threaded binary tree construction for star graphs based on distance levels and characterizes the distribution of distances in efficient dominating sets.

Findings

Distribution of distances in star graphs is established

A threaded binary tree representation of star graphs is constructed

Distances of efficient dominating sets are characterized

Abstract

The distribution of distances in the star graph $ST_n$, ($1<n\in\Z$), is established, and subsequently a threaded binary tree is obtained that realizes an orientation of $ST_n$ 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 $ST_n$ are determined.