Computing the second and third systoles of a combinatorial surface

TL;DR

This paper presents algorithms for efficiently computing the second and third shortest non-trivial closed walks on a surface-embedded graph, advancing topological graph algorithms with improved runtimes.

Contribution

It introduces algorithms with optimal runtimes for finding the second and third shortest homotopically non-trivial closed walks on a surface.

Findings

Algorithms run in O(n^2 log n) and O(n^3) time for second and third shortest walks.

Reduces runtime to O(n log n) for fixed genus and boundaries.

Provides methods to compute shortest essential arcs between vertices.

Abstract

Given a weighted, undirected graph $G$ cellularly embedded on a topological surface $S$, we describe algorithms to compute the second shortest and third shortest closed walks of $G$ that are neither homotopically trivial in $S$ nor homotopic to the shortest non-trivial closed walk or to each other. Our algorithms run in $O(n^2\log n)$ time for the second shortest walk and in $O(n^3)$ time for the third shortest walk. We also show how to reduce the running time for the second shortest homotopically non-trivial closed walk to $O(n\log n)$ when both the genus and the number of boundaries are fixed. Our algorithms rely on a careful analysis of the configurations of the first three shortest homotopically non-trivial curves in $S$. As an intermediate step, we also describe how to compute a shortest essential arc between \emph{one} pair of vertices or between \emph{all} pairs of vertices of a given boundary component of $S$ in $O(n^2)$ time or $O(n^3)$ time, respectively.