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. 2023 Jan 12:1–44. Online ahead of print. doi: 10.1007/s00453-022-01089-6

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© The Author(s) 2023

Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 5 — This figure shows the cycle formed by adding $e_{1}$ to the SMRCST. The cycle is of length d and contains the nodes $v_{1}, \dots, v_{d}$ . Every other node is contained in one of the subtrees rooted in one of the nodes on the cycle. These subtrees are represented in yellow. The number of nodes contained in the subtree rooted in $v_{i}$ is $x_{i}$ . Middle and right: the cycle for d being odd or even, respectively, and the two special edges $e_{2}$ and $e_{3}$ (Color figure online)