Skip to main content
Springer Nature Link
Log in

Circular Convex Bipartite Graphs: Feedback Vertex Set

  • Conference paper
Combinatorial Optimization and Applications (COCOA 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8287))

Abstract

A feedback vertex set is a subset of vertices, such that the removal of this subset renders the remaining graph cycle-free. The weight of a feedback vertex set is the sum of weights of its vertices. Finding a minimum weighted feedback vertex set is tractable for convex bipartite graphs, but \(\mathcal{NP}\)-complete even for unweighted bipartite graphs. In a circular convex (convex, respectively) bipartite graph, there is a circular (linear, respectively) ordering defined on one class of vertices, such that for every vertex in another class, the neighborhood of this vertex is a circular arc (an interval, respectively). The minimum weighted feedback vertex set problem is shown tractable for circular convex bipartite graphs in this paper, by making a Cook reduction (i.e. polynomial time Turing reduction) for this problem from circular convex bipartite graphs to convex bipartite graphs.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Bao, F.S., Zhang, Y.: A review of tree convex sets test. Computational Intelligence 28(3), 358–372 (2012); Previous version: A survey of tree convex sets test. arXiv.0906.0205 (2009)

    Google Scholar 

  2. Becker, A., Bar-Yehuda, R., Geiger, D.: Randomized Algorithms for the Loop Cutset Problem. J. Artif. Intell. Res. 12, 219–234 (2000)

    MathSciNet  MATH  Google Scholar 

  3. Brandstad, A., Le, V.B., Spinrad, J.P.: Graph Classes - A Survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)

    Book  Google Scholar 

  4. Cao, Y., Chen, J., Liu, Y.: On Feedback Vertex Set: New Measure and New Structures. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 93–104. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  5. Dom, M.: Algorithmic aspects of the consecutive ones property. Bulletin of the EATCS 98, 27–59 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Damaschke, P., Muller, H., Kratsch, D.: Domination in Convex and Chordal Bipartite Graphs. Inform. Proc. Lett. 36, 231–236 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fomin, F.V., Gaspers, S., Pyatkin, A., Razgon, I.: On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms. Algorithmica 52(2), 293–307 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Festa, P., Pardalos, P.M., Resende, M.G.C.: Feedback set problems. In: Handbook of Combinatorial Optimization, (suppl. vol. A), pp. 209–258. Kluwer Academic Publishers (1999)

    Google Scholar 

  9. Fomin, F.V., Villanger, Y.: Finding Induced Subgraphs via Minimal Triangulations. In: Proc. of STACS, pp. 383–394 (2010)

    Google Scholar 

  10. Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company (1979)

    Google Scholar 

  11. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)

    MATH  Google Scholar 

  12. Golumbic, M.C., Goss, C.F.: Perfect elimination and chordal bipartite graphs. J. Graph Theory. 2, 155–163 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grover, F.: Maximum matching in a convex bipartite graph. Nav. Res. Logist. Q. 14, 313–316 (1967)

    Article  Google Scholar 

  14. Guo, J.: Undirected feedback vertex set. Encyclopedia of Algorithms, 995–996 (2008)

    Google Scholar 

  15. Hung, R.-W.: Linear-time algorithm for the paired-domination problem in convex bipartite graphs. Theory Comput. Syst. 50, 721–738 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jiang, W., Liu, T., Ren, T., Xu, K.: Two hardness results on feedback vertex sets. In: Atallah, M., Li, X.-Y., Zhu, B. (eds.) FAW-AAIM 2011. LNCS, vol. 6681, pp. 233–243. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  17. Jiang, W., Liu, T., Wang, C., Xu, K.: Feedback vertex sets on restricted bipartite graphs. Theor. Comput. Sci. (in press, 2013), doi: 10.1016/j.tcs.2012.12.021

    Google Scholar 

  18. Jiang, W., Liu, T., Xu, K.: Tractable feedback vertex sets in restricted bipartite graphs. In: Wang, W., Zhu, X., Du, D.-Z. (eds.) COCOA 2011. LNCS, vol. 6831, pp. 424–434. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  19. Karp, R.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

    Chapter  Google Scholar 

  20. Kloks, T., Liu, C.H., Pon, S.H.: Feedback vertex set on chordal bipartite graphs. arXiv:1104.3915 (2011)

    Google Scholar 

  21. Kloks, T., Wang, Y.L.: Advances in graph algorithms. Manuscipt of a book (2013)

    Google Scholar 

  22. Liang, Y.D., Blum, N.: Circular convex bipartite graphs: maximum matching and Hamiltonian circuits. Inf. Process. Lett. 56, 215–219 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liang, Y.D., Chang, M.S.: Minimum feedback vertex sets in cocomparability graphs and convex bipartite graphs. Acta Informatica 34, 337–346 (1997)

    Article  MathSciNet  Google Scholar 

  24. Lu, M., Liu, T., Xu, K.: Independent Domination: Reductions from Circular- and Triad-Convex Bipartite Graphs to Convex Bipartite Graphs. In: Fellows, M., Tan, X., Zhu, B. (eds.) FAW-AAIM 2013. LNCS, vol. 7924, pp. 142–152. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  25. Lu, Z., Liu, T., Xu, K.: Tractable Connected Domination for Restricted Bipartite Graphs (Extended Abstract). In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 721–728. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  26. Madelaine, F.R., Stewart, I.A.: Improved upper and lower bounds on the feedback vertex numbers of grids and butterflies. Discrete Math. 308, 4144–4164 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Song, Y., Liu, T., Xu, K.: Independent domination on tree convex bipartite graphs. In: Snoeyink, J., Lu, P., Su, K., Wang, L. (eds.) AAIM 2012 and FAW 2012. LNCS, vol. 7285, pp. 129–138. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  28. Wang, C., Liu, T., Jiang, W., Xu, K.: Feedback vertex sets on tree convex bipartite graphs. In: Lin, G. (ed.) COCOA 2012. LNCS, vol. 7402, pp. 95–102. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  29. Wang, F.H., Wang, Y.L., Chang, J.M.: Feedback vertex sets in star graphs. Inform. Process. Lett. 89(4), 203–208 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yannakakis, M.: Node-deletion problem on bipartite graphs. SIAM J. Comput. 10, 310–327 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhou, H.: The feedback vertex set problem: a spin glass approach. arXiv:1307.6948 (2013)

    Google Scholar 

  32. Van Zuylen, A.: Linear programming based approximation algorithms for feedback set problems in bipartite tournaments. Theor. Comput. Sci. (in press)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of High Confidence Software Technologies, Ministry of Education, Institute of Software, School of Electronic Engineering and Computer Science, Peking University, Beijing, 100871, China

    Zhao Lu, Min Lu & Tian Liu

  2. National Lab of Software Development Environment, Beihang University, Beijing, 100191, China

    Ke Xu

Authors
  1. Zhao Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Min Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tian Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ke Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland

    Peter Widmayer

  2. State Key Lab for Manufacturing Systems Engineering, 710049, Xi’an, China

    Yinfeng Xu

  3. Department of Computer Science, Montana State University, 59717-3880, Bozeman, MT, USA

    Binhai Zhu

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this paper

Cite this paper

Lu, Z., Lu, M., Liu, T., Xu, K. (2013). Circular Convex Bipartite Graphs: Feedback Vertex Set. In: Widmayer, P., Xu, Y., Zhu, B. (eds) Combinatorial Optimization and Applications. COCOA 2013. Lecture Notes in Computer Science, vol 8287. Springer, Cham. https://doi.org/10.1007/978-3-319-03780-6_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-03780-6_24

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-03779-0

  • Online ISBN: 978-3-319-03780-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics