On the complexity of the planar edge-disjoint paths problem with terminals on the outer boundary

It is shown that both the undirected and the directed edge-disjoint paths problem are NP-complete, if the supply graph is planar and all edges of the demand graph are incident with vertices lying on the outer boundary of the supply graph. In the directed case, the problem remains NP-complete, if in addition the supply graph is acyclic. The undirected case solves open problem no. 56 of A. Schrijver’s book Combinatorial Optimization.