Tree t-spanners in Outerplanar Graphs via Supply Demand Partition

TL;DR

This paper introduces a linear-time algorithm for finding tree t-spanners in outerplanar graphs by reducing the problem to supply-demand tree partition, enabling efficient computation of the minimal t value.

Contribution

It presents a novel linear-time reduction from tree t-spanner problem in outerplanar graphs to supply-demand tree partition, and an algorithm to determine the minimal t.

Findings

Linear-time algorithm for tree t-spanner in outerplanar graphs

Reduction from tree t-spanner to supply-demand tree partition

Efficient computation of minimal t in O(n log n) time

Abstract

A tree t-spanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the tree t-spanner problem is to compute a tree t-spanner of G if one exists. This decision problem is known to be NP-complete even in the restricted class of unweighted planar graphs. We present a linear-time reduction from tree t-spanner in outerplanar graphs to the supply-demand tree partition problem. Based on this reduction, we obtain a linear-time algorithm to solve tree t-spanner in outerplanar graphs. Consequently, we show that the minimum value of t for which an input outerplanar graph on n vertices has a tree t-spanner can be found in O(n log n) time.