Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Frank: Edge-disjoint paths in planar graphs, Journal of Combinatorial Theory, Series B 39 (1985), 164–178.
M. R. Kramer and J. van Leeuwen: The complexity of wire-routing and finding minimum area layouts for arbitrary VLSI circuits, in: VLSI-Theory (F. P. Preparata, ed.), Advances in Computing Research, Volume 2, JAI Press, Greenwich, Connecticut, (1984), 129–146.
M. Middendorf and F. Pfeiffer: On the complexity of the disjoint paths problem, Combinatorica 13(1) (1993), 97–107.
H. Okamura and P. D. Seymour: Multicommodity flows in planar graphs, Journal of Combinatorial Theory, Series B 31 (1981), 75–81.
A. Schrijver: Combinatorial Optimization — Polyhedra and Efficiency, Springer, Berlin, 2003.
J. Vygen: NP-completeness of some edge-disjoint paths problems, Discrete Applied Mathematics 61 (1995), 83–90.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schwärzler, W. On the complexity of the planar edge-disjoint paths problem with terminals on the outer boundary. Combinatorica 29, 121–126 (2009). https://doi.org/10.1007/s00493-009-2407-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-009-2407-4