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 [1] and later W [1]-hard by Pilipczuk and Wahlstr\"om [7], parametrized by k. On the other hand, there is an XP algorithm by Cygan et al. [3] and a polynomial time algorithm for graphs without directed edges by Hassin and Megiddo [5]. Chitnis and Feldmann [2] showed W [1]-hardness of the problem for graphs of genus 1. We consider a further restriction to planar graphs and show NP -completeness.
@article{arxiv.1804.07496,
title = {Planar Steiner Orientation is NP-complete},
author = {Moritz Beck and Johannes Blum and Myroslav Kryven and Andre Löffler and Johannes Zink},
journal= {arXiv preprint arXiv:1804.07496},
year = {2018}
}