Many applications in graph theory are motivated by routing or flow problems. Among these problems is Steiner Orientation: given a mixed graph G (having directed and undirected edges) and a set T of k terminal pairs in G, is there an orientation of the undirected edges in G such that there is a directed path for every terminal pair in T? This problem was shown to be NP-complete by Arkin and Hassin and later W[1]-hard by Pilipczuk and Wahlstrom, parametrized by k. On the other hand, there is an XP algorithm by Cygan et al. and a polynomial time algorithm for graphs without directed edges by Hassin and Megiddo. Chitnis and Feldmann showed W[1]-hardness of the problem for graphs of genus 1. We consider a further restriction to planar graphs and show NP-completeness.
Paper by Moritz Beck, Johannes Blum, Myroslav Kryven, Andre Loffler, and Johannes Zink. https://arxiv.org/pdf/1804.07496.pdf