Algorithms for Diameters of Unicycle Graphs and Diameter-Optimally Augmenting Trees

TL;DR

This paper introduces an optimal O(n) algorithm for computing the diameter of unicycle graphs and applies it to efficiently minimize the diameter when augmenting trees with shortcuts, improving previous methods.

Contribution

It presents the first linear-time algorithm for unicycle graph diameter computation and a more efficient method for diameter-optimally augmenting trees.

Findings

O(n) time algorithm for unicycle graph diameter

Improved O(n^2 log n) algorithm for diameter minimization in tree augmentation

Enhanced efficiency over previous algorithms in both time and space complexity

Abstract

We consider the problem of computing the diameter of a unicycle graph (i.e., a graph with a unique cycle). We present an O(n) time algorithm for the problem, where n is the number of vertices of the graph. This improves the previous best O(n \log n) time solution [Oh and Ahn, ISAAC 2016]. Using this algorithm as a subroutine, we solve the problem of adding a shortcut to a tree so that the diameter of the new graph (which is a unicycle graph) is minimized; our algorithm takes O(n^2 \log n) time and O(n) space. The previous best algorithms solve the problem in O(n^2 \log^3 n) time and O(n) space [Oh and Ahn, ISAAC 2016], or in O(n^2) time and O(n^2) space [Bil\`o, ISAAC 2018].