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arXiv · 2309.10122

Graph Threading

Erik D. Demaine·Yael Kirkpatrick·Rebecca Lin
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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.

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Erik D. Demaine, Yael Kirkpatrick, Rebecca Lin. 2023-09-18. Graph Threading. https://doi.org/10.4230/lipics.itcs.2024.38

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