Some Preliminary Result About the Inset Edge and Average Distance of Trees

TL;DR

This paper introduces tools for analyzing the impact of adding inset edges to trees, enabling more efficient algorithms and tighter bounds on average distance changes, especially for the case of a single inset edge.

Contribution

It develops new analytical tools for trees that improve understanding and computation of average distance changes when inset edges are added.

Findings

Provided a tight bound for average distance change in trees with inset edges.

Developed algorithms avoiding distance matrix calculations.

Analyzed the problem for the case of a single inset edge.

Abstract

An added edge to a graph is called an inset edge. Predicting k inset edges which minimize the average distance of a graph is known to be NP-Hard. However, when k = 1 the complexity of the problem is polynomial. In this paper, some tools for a precise analysis of the problem for the trees are established. Using the tools, we can avoid using the distance matrix. This leads to more efficient algorithms and a better analysis of the problem. Several applications of the tools as well as a tight bound for the change of average distance when an inset edge is added to a tree are presented.