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.
@article{arxiv.2410.06793,
title = {A Polynomial Time Algorithm for Steiner Tree when Terminals Avoid a $K_4$-Minor},
author = {Carla Groenland and Jesper Nederlof and Tomohiro Koana},
journal= {arXiv preprint arXiv:2410.06793},
year = {2024}
}