  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.
