Computing the Girth of a Planar Graph in Linear Time

TL;DR

This paper presents a groundbreaking linear-time algorithm for computing the girth of an unweighted undirected planar graph, significantly improving previous algorithms with higher polynomial or logarithmic complexities.

Contribution

We introduce the first linear-time algorithm for determining the girth of planar graphs, surpassing prior methods with higher computational complexities.

Findings

Girth can be computed in O(n) time for planar graphs.

Previous algorithms had higher polynomial or logarithmic complexities.

The new algorithm achieves optimal linear time complexity.

Abstract

The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first non-trivial algorithm for the problem, given by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further reduced the running time to O(n log n). In this paper, we solve the problem in O(n) time.