We study a special case of the Steiner Tree problem in which the input graph does not have a minor model of a complete graph on 4 vertices for which all branch sets contain a terminal. We show that this problem can be solved in O(n⁴) time, where n denotes the number of vertices in the input graph. This generalizes a seminal paper by Erickson et al. [Math. Oper. Res., 1987] that solves Steiner tree on planar graphs with all terminals on one face in polynomial time.
@InProceedings{groenland_et_al:LIPIcs.IPEC.2024.12, author = {Groenland, Carla and Nederlof, Jesper and Koana, Tomohiro}, title = {{A Polynomial Time Algorithm for Steiner Tree When Terminals Avoid a Rooted K₄-Minor}}, booktitle = {19th International Symposium on Parameterized and Exact Computation (IPEC 2024)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-353-9}, ISSN = {1868-8969}, year = {2024}, volume = {321}, editor = {Bonnet, \'{E}douard and Rz\k{a}\.{z}ewski, Pawe{\l}}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2024.12}, URN = {urn:nbn:de:0030-drops-222387}, doi = {10.4230/LIPIcs.IPEC.2024.12}, annote = {Keywords: Steiner tree, rooted minor} }
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