Graph Threading

TL;DR

This paper introduces a polynomial-time algorithm for finding minimal-length threadings through tubes that form a specified connected graph, with applications inspired by artistic threading practices.

Contribution

It presents a novel polynomial-time algorithm for the graph threading problem, including tight bounds and efficient solutions for special cases.

Findings

Polynomial-time algorithm for minimum-length threading

Tight bounds on threading length and edge visits

Efficient solutions for cubic graphs and limited edge visits

Abstract

Inspired by artistic practices such as beadwork and himmeli, we study the problem of threading a single string through a set of tubes, so that pulling the string forms a desired graph. More precisely, given a connected graph (where edges represent tubes and vertices represent junctions where they meet), we give a polynomial-time algorithm to find a minimum-length closed walk (representing a threading of string) that induces a connected graph of string at every junction. The algorithm is based on a surprising reduction to minimum-weight perfect matching. Along the way, we give tight worst-case bounds on the length of the optimal threading and on the maximum number of times this threading can visit a single edge. We also give more efficient solutions to two special cases: cubic graphs and the case when each edge can be visited at most twice.