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How Close Is a Tree to a Euclidean Minimum Spanning Tree?

Authors Todor Antić , Jiří Fiala , Jelena Glišić , Grzegorz Gutowski , Konstanty Junosza-Szaniawski , Jan Kratochvíl , Giuseppe Liotta , Morteza Saghafian , Maria Saumell , Krisztina Szilágyi , Pavel Valtr

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Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
Keywords
  • Straight-line Drawing of Tree
  • Euclidean Minimum Spanning Tree
  • EMST-Drawability
  • Caterpillar
  • Graph Drawing Algorithm

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Abstract

Let Γ be a straight-line crossing-free drawing of a tree T. A bad pair in Γ is a pair of non-adjacent vertices of T whose Euclidean distance in Γ is smaller than the length of the longest edge in the path connecting them in Γ. When Γ has no bad pairs, Γ is a Euclidean Minimum Spanning Tree of its vertex set (or EMST-drawing for short). Deciding whether a tree of maximum degree at most six admits an EMST-drawing is known to be NP-hard. In contrast, we characterize those caterpillars that admit an EMST-drawing. The characterization gives rise to a linear-time algorithm that decides if a caterpillar admits an EMST-drawing, and in the affirmative case, computes such a drawing. For caterpillars of maximum degree six, we further present a linear-time algorithm to compute a crossing-free straight-line drawing with the minimum number of bad pairs. For n-vertex trees with maximum vertex degree Δ, we prove the Δ²nlog n upper bound on the minimum number of bad pairs. In the special case of stars, we construct a drawing with the minimum number of bad pairs.

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Todor Antić, Jiří Fiala, Jelena Glišić, Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Jan Kratochvíl, Giuseppe Liotta, Morteza Saghafian, Maria Saumell, Krisztina Szilágyi, and Pavel Valtr. How Close Is a Tree to a Euclidean Minimum Spanning Tree?. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.9

Author Details

Todor Antić
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Jiří Fiala
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Jelena Glišić
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Grzegorz Gutowski
  • Institute of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Konstanty Junosza-Szaniawski
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland
Jan Kratochvíl
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Giuseppe Liotta
  • Università degli Studi di Perugia, Perugia, Italy
Morteza Saghafian
  • Algorithm and Complexity Group, TU Wien, Vienna, Austria
Maria Saumell
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
Krisztina Szilágyi
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
Pavel Valtr
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

Funding

  • Antić, Todor: Supported by grant no. 23-04949X of the Czech Science Foundation (GAČR) and grant SVV–2025–260822 of GA UK.
  • Fiala, Jiří: Supported by grant no. 25-16847S of the Czech Science Foundation (GAČR).
  • Glišić, Jelena: Supported by grant no. 23-04949X of the Czech Science Foundation (GAČR) and grant SVV–2025–260822 of GA UK.
  • Gutowski, Grzegorz: Partially supported by the National Science Centre, Poland, grant no. 2023/51/B/ST6/02833.
  • Kratochvíl, Jan: Supported by grant no. 23-04949X of the Czech Science Foundation (GAČR).
  • Saumell, Maria: Supported by grant no. 23-04949X of the Czech Science Foundation (GAČR).
  • Szilágyi, Krisztina: Supported under the project Robotics and advanced industrial production (reg. no. CZ.02.01.01/00/22_008/0004590) and CTU Global Postdoc Fellowship Program.
  • Valtr, Pavel: Supported by grant no. 23-04949X of the Czech Science Foundation (GAČR).

Acknowledgements

This work was started at the workshop HOMONOLO 2025 in Nová Louka. The authors thank the organizers and all other participants for a productive and inspiring atmosphere. We also thank anonymous reviewers for their helpful comments.

References

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