Skip to main content
Log in

The complexity of induced minors and related problems

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

The computational complexity of a number of problems concerning induced structures in graphs is studied, and compared with the complexity of corresponding problems concerning non-induced structures. The effect on these problems of restricting the input to planar graphs is also considered. The principal results include: (1) Induced Maximum Matching and Induced Directed Path are NP-complete for planar graphs, (2) for every fixed graphH, InducedH-Minor Testing can be accomplished for planar graphs in time0(n), and (3) there are graphsH for which InducedH-Minor Testing is NP-complete for unrestricted input. Some useful structural theorems concerning induced minors are presented, including a bound on the treewidth of planar graphs that exclude a planar induced minor.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bodlaender, H., A linear time algorithm for finding tree-decompositions of small treewidth,Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, 1993, pp. 226–234.

  2. Cameron, K., Induced matchings,Discrete Appl. Math. 24 (1989), 97–102.

    Google Scholar 

  3. Courcelle, B., The monadic second-order logic of graphs, I: Recognizable sets of finite graphs,Inform. and Control (to appear).

  4. Edmonds, J., Maximum matching and a polyhedron with 0, 1 vertices,J. Res. Nat. Bur. Standards 69B (1965), 125–130.

    Google Scholar 

  5. Fellows, M. R., The Robertson-Seymour theorems: A survey of applications,Contemp. Math. 89 (1989), 1–18.

    Google Scholar 

  6. Fortune, S., Hopcroft, J. E., and Wyllie, J., The directed subgraph homeomorphism problem,J. Theoret. Comput. Sci. 10 (1980), 111–121.

    Google Scholar 

  7. Garey, M. R., and Johnson, D. S.,Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, New York, 1979.

    Google Scholar 

  8. Karp, R. M., On the complexity of combinatorial problems,Networks 5 (1975), 45–68.

    Google Scholar 

  9. Kratochvil, J., Lubiw, A., and Nešetřil, J., Noncrossing subgraphs in topological layouts,SIAM J. Discrete Math. 4 (1991), 223–244.

    Google Scholar 

  10. Lichtenstein, D., Planar formulae and their uses,SIAM J. Comput. 11 (1982), 329–343.

    Google Scholar 

  11. Lubiw, A., A note on odd/even cycles,Discrete Appl. Math. 22 (1988/9), 87–92.

    Google Scholar 

  12. Matoušek, J., Nešetřil, J., and Thomas, R., On polynomial time decidability of induced minor closed classes,Comment. Math. Univ. Carolin. 29 (1988), 703–710.

    Google Scholar 

  13. McDiarmid, C, Reed, B., Schrijver, A., and Shepherd, B., Induced circuits in planar graphs, Preprint.

  14. McDiarmid, C., Reed, B., Schrijver, A., and Shepherd, B., Non-interfering paths in planar digraphs, Preprint.

  15. Moret, B., Planar not-all-equal 3sat is inP, SIGACT News (1989).

  16. Robertson, N, and Seymour, P. D., Graph minors I-XVIII.

  17. Schrijver, A., Disjoint homotopic paths and trees in a planar graph,Discrete Comput. Geom. 6 (1991), 527–574.

    Google Scholar 

  18. Seymour, P. D., Disjoint paths in graphs,Discrete Math. 29 (1980), 293–309.

    Google Scholar 

  19. Shiloach, Y., A polynomial solution to the undirected two paths problem,J. Assoc. Comput. Mach. 27 (1980), 445–456.

    Google Scholar 

  20. Thomas, R., Graphs without K4 and well-quasi-ordering,J. Combin. Theory Ser. B 38 (1985), 445–456.

    Google Scholar 

  21. Thomassen, C., 2-linked graphs,European J. Combin. 29 (1980), 371–378.

    Google Scholar 

  22. Valiant, L. G., Universality considerations in VLSI circuits,IEEE Trans. Comput. 30 (1981), 135–140.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by C. H. Papadimitriou.

The research of the first author was supported by the U.S. Office of Naval Research under Contract N00014-88-K-0456, by the U.S. National Science Foundation under Grant MIP-8603879, and by the National Science and Engineering Research Council of Canada. The second author acknowledges the support of the U.S. Office of Naval Research when visiting the University of Idaho in spring 1990. Some results were also obtained during a visit to the University of Cologne in fall 1990.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fellows, M.R., Kratochvil, J., Middendorf, M. et al. The complexity of induced minors and related problems. Algorithmica 13, 266–282 (1995). https://doi.org/10.1007/BF01190507

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01190507

Key words

Navigation