conference paper published yes Sebastiano Cultrera di Montesano author 34D2A09C-F248-11E8-B48F-1D18A9856A870000-0001-6249-0832 Ondrej Draganov author 2B23F01E-F248-11E8-B48F-1D18A9856A870000-0003-0464-3823 Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 Morteza Saghafian author f86f7148-b140-11ec-9577-95435b8df824 HeEd department GD: Graph Drawing and Network Visualization Alpha Shape Theory Extended project Mathematics, Computer Science project Persistence and stability of geometric complexes project Given a finite set, A ⊆ ℝ², and a subset, B ⊆ A, the MST-ratio is the combined length of the minimum spanning trees of B and A⧵B divided by the length of the minimum spanning tree of A. The question of the supremum, over all sets A, of the maximum, over all subsets B, is related to the Steiner ratio, and we prove this sup-max is between 2.154 and 2.427. Restricting ourselves to 2-dimensional lattices, we prove that the sup-max is 2, while the inf-max is 1.25. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than 1.25. https://research-explorer.ista.ac.at/download/18556/18560/2024_LIPIcs_CultreradiMontesano.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Schloss Dagstuhl - Leibniz-Zentrum für Informatik2024Vienna, Austria eng 1868-8969 9783959773430 2403.10204 00154027840000110.4230/LIPIcs.GD.2024.3 320 Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2024. The Euclidean MST-ratio for bi-colored lattices. 32nd International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network Visualization, LIPIcs, vol. 320, 3. S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, in:, 32nd International Symposium on Graph Drawing and Network Visualization, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. Cultrera di Montesano, Sebastiano, et al. “The Euclidean MST-Ratio for Bi-Colored Lattices.” <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, vol. 320, 3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href="https://doi.org/10.4230/LIPIcs.GD.2024.3">10.4230/LIPIcs.GD.2024.3</a>. Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. The Euclidean MST-ratio for bi-colored lattices. In: <i>32nd International Symposium on Graph Drawing and Network Visualization</i>. Vol 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href="https://doi.org/10.4230/LIPIcs.GD.2024.3">10.4230/LIPIcs.GD.2024.3</a> Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “The Euclidean MST-Ratio for Bi-Colored Lattices.” In <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, Vol. 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href="https://doi.org/10.4230/LIPIcs.GD.2024.3">https://doi.org/10.4230/LIPIcs.GD.2024.3</a>. Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2024). The Euclidean MST-ratio for bi-colored lattices. In <i>32nd International Symposium on Graph Drawing and Network Visualization</i> (Vol. 320). Vienna, Austria: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href="https://doi.org/10.4230/LIPIcs.GD.2024.3">https://doi.org/10.4230/LIPIcs.GD.2024.3</a> S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “The Euclidean MST-ratio for bi-colored lattices,” in <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, Vienna, Austria, 2024, vol. 320. 185562024-11-17T23:01:47Z2025-12-02T13:50:50Z