Skip to main content
SpringerLink
Log in
Book cover

International Conference on Theory and Applications of Models of Computation

TAMC 2022: Theory and Applications of Models of Computation pp 103–115Cite as

Analyzing the 3-path Vertex Cover Problem in Planar Bipartite Graphs

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13571))

Abstract

Let \(G=(V,E)\) be a simple graph. A set \(C \subseteq V\) is called a k-path vertex cover of G, if each k-path in G contains at least one vertex from C. In the k-path vertex cover problem, we are given a graph G and asked to find a k-path vertex cover of minimum cardinality. For \(k=3\), the problem becomes the well-known 3-path vertex cover (3PVC) problem, which has been widely studied, as per the literature. In this paper, we focus on the 3PVC problem in planar bipartite (pipartite) graphs for the most part. We first show that the 3PVC problem is NP-hard, even in pipartite graphs in which the degree of all vertices is bounded by 4. We then show that the 3PVC problem on this class of graphs admits a linear time 1.5-approximation algorithm. Finally, we show that the 3PVC problem is APX-complete in bipartite graphs. The last result is particularly interesting, since the vertex cover problem in bipartite graphs is solvable in polynomial time.

This research was supported in part by the Air-Force Office of Scientific Research through Grant FA9550-19–1-0177 and in part by the Air-Force Research Laboratory, Rome through Contract FA8750-17-S-7007.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Brešar, B., Kardoš, F., Katrenič, J., Semanišin, G.: Minimum k-path vertex cover. Discrete Appl. Math. 159(12), 1189–1195 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1990)

    Google Scholar 

  3. Devi, N.S., Mane, A.C., Mishra, S.: Computational complexity of minimum P4 vertex cover problem for regular and K1, 4-free graphs. Discrete Appl. Math. 184, 114–121 (2015)

    Google Scholar 

  4. Du, D., Ko, K.-I., Hu, X., et al.: Design and Analysis of Approximation Algorithms, vol. 62. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-1701-9

  5. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. TTCSAES, Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16533-7

    Book  MATH  Google Scholar 

  6. Gollmann, D.: Protocol analysis for concrete environments. In: Moreno Díaz, R., Pichler, F., Quesada Arencibia, A. (eds.) EUROCAST 2005. LNCS, vol. 3643, pp. 365–372. Springer, Heidelberg (2005). https://doi.org/10.1007/11556985_47

    Chapter  Google Scholar 

  7. Kardoš, F., Katrenič, J., Schiermeyer, I.: On computing the minimum 3-path vertex cover and dissociation number of graphs. Theoret. Comput. Sci. 412(50), 7009–7017 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  9. Kumar, M., Mishra, S., Devi, N.S., Saurabh, S.: Approximation algorithms for node deletion problems on bipartite graphs with finite forbidden subgraph characterization. Theoret. Comput. Sci. 526, 90–96 (2014)

    Google Scholar 

  10. Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Novotnỳ, M.: Design and analysis of a generalized canvas protocol. In: IFIP International Workshop on Information Security Theory and Practices, pp. 106–121 (2010)

    Google Scholar 

  12. Tu, J., Shi, Y.: An efficient polynomial time approximation scheme for the vertex cover \(p_3\) problem on planar graphs. Discussiones Math. Graph Theory 39(1), 2019

    Google Scholar 

  13. Jianhua, T., Yang, F.: The vertex cover P3 problem in cubic graphs. Inf. Process. Lett. 113(13), 481–485 (2013)

    Article  MATH  Google Scholar 

  14. Jianhua, T., Zhou, W.: A factor 2 approximation algorithm for the vertex cover P3 problem. Inf. Process. Lett. 111(14), 683–686 (2011)

    Article  MATH  Google Scholar 

  15. Jianhua, T., Zhou, W.: A primal-dual approximation algorithm for the vertex cover P3 problem. Theoret. Comput. Sci. 412(50), 7044–7048 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, A., Chen, Y., Chen, Z.-Z., Lin, G.: Improved approximation algorithms for path vertex covers in regular graphs. Algorithmica 82(10), 3041–3064 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LDCSEE, West Virginia University, Morgantown, WV, USA

    Sangram K. Jena & K. Subramani

Authors
  1. Sangram K. Jena
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Subramani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Subramani .

Editor information

Editors and Affiliations

  1. University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. University of New Brunswick, Fredericton, NB, Canada

    Donglei Du

  3. Tianjin University of Technology, Tianjin, China

    Chenchen Wu

  4. Beijing University of Technology, Beijing, China

    Dachuan Xu

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Jena, S.K., Subramani, K. (2022). Analyzing the 3-path Vertex Cover Problem in Planar Bipartite Graphs. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Theory and Applications of Models of Computation. TAMC 2022. Lecture Notes in Computer Science, vol 13571. Springer, Cham. https://doi.org/10.1007/978-3-031-20350-3_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-20350-3_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-20349-7

  • Online ISBN: 978-3-031-20350-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation