Abstract
In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graphG embedded in ℝ2, a subset {I 1, …,I p} of the faces ofG, and pathsC 1, …,C k inG, with endpoints on the boundary ofI 1 ∪ … ∪I p; find: pairwise disjoint simple pathsP 1, …,P k inG so that, for eachi=1, …,k, P i is homotopic toC i in the space ℝ2\(I 1 ∪ … ∪I p).
Moreover, we prove a theorem characterizing the existence of a solution to this problem. Finally, we extend the algorithm to disjoint homotopic trees. As a corollary we derive that, for each fixedp, there exists a polynormial-time algorithm for the problem:given: a planar graphG embedded in ℝ2 and pairwise disjoint setsW 1, …,W k of vertices, which can be covered by the boundaries of at mostp faces ofG;find: pairwise vertex-disjoint subtreesT 1, …,T k ofG whereT i (i=1, …, k).
Article PDF
Similar content being viewed by others
References
R. Cole and A. Siegel, River routing every which way, but loose,Proceedings of the 25th Annual Symposium on Foundations of Computer Science, 1984, pp. 65–73.
A. Frank and A. Schrijver, Vertex-disjoint simple paths of given homotopy in a planar graph, in:Polyhedral Combinatorics (W. Cook and P. D. Seymour, eds), American Mathematical Society, Providence, RI, 1990, pp. 139–161.
C. E. Leiserson and F. M. Maley, Algorithms for routing and testing routability of planar VLSI layouts,Proceedings of the 17th Annual ACM Symposium on Theory of Computing, 1985, pp. 69–78.
W. S. Massey,Algebraic Topology: An Introduction, Springer-Verlag, New York, 1967.
R. Y. Pinter, River routing: methodology and analysis,Third CalTech Conference on Very Large Scale Integration (R. Bryant, ed.), Springer-Verlag, Berlin, 1983, pp. 141–163.
N. Robertson and P. D. Seymour, Graph minors, VI. Disjoint paths across a disc,J. Combin. Theory Ser. B 41 (1986), 115–138.
N. Robertson and P. D. Seymour, Graph minors, VII. Disjoint paths on a surface,J. Combin. Theory Ser. B 45 (1988), 212–254.
N. Robertson and P. D. Seymour, Graph minors, XV, to appear.
A. Schrijver, Disjoint circuits of prescribed homotopies in a graph on a compact surface,J. Combin. Theory Ser. B 51 (1991), 127–159.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schrijver, A. Disjoint homotopic paths and trees in a planar graph. Discrete Comput Geom 6, 527–574 (1991). https://doi.org/10.1007/BF02574704
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02574704