Skip to main content
SpringerLink
Log in

On the Complexity of Minimum Maximal Uniquely Restricted Matching

  • Conference paper
  • First Online:
Combinatorial Optimization and Applications (COCOA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12577))

  • 630 Accesses

Abstract

A subset \(M\subseteq E\) of edges of a graph \(G=(V,E)\) is called a matching if no two edges of M share a common vertex. A matching M in a graph G is called a uniquely restricted matching if G[V(M)], the subgraph of G induced by the M-saturated vertices of G, contains exactly one perfect matching. A uniquely restricted matching M is maximal if M is not properly contained in any other uniquely restricted matching of G. Given a graph G, the Min-Max-UR Matching problem asks to find a maximal uniquely restricted matching of minimum cardinality in G. In general, the decision version of the Min-Max-UR Matching problem is known to be \({\mathsf {NP}}\)-complete for general graphs and remains so even for bipartite graphs. In this paper, we strengthen this result by proving that this problem remains \({\mathsf {NP}}\)-complete for chordal bipartite graphs and chordal graphs. On the positive side, we prove that the Min-Max-UR Matching problem is polynomial time solvable for bipartite permutation graphs and proper interval graphs. Finally, we show that the Min-Max-UR Matching problem is \(\mathsf {APX}\)-complete for bounded degree graphs.

J. Chaudhary—The author has been supported by the Department of Science and Technology through INSPIRE Fellowship for this research.

B. S. Panda—The author wants to thank the SERB, Department of science and technology for their support vide Diary No. SERB/F/12949/2018-2019.

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

Access this chapter

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

Institutional subscriptions

References

  1. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Spaccamela, A.M., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  2. Baste, J., Rautenbach, D., Sau, I.: Uniquely restricted matchings and edge colorings. In: Bodlaender, H.L., Woeginger, G.J. (eds.) WG 2017. LNCS, vol. 10520, pp. 100–112. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68705-6_8

    Chapter  Google Scholar 

  3. Francis, M.C., Jacob, D., Jana, S.: Uniquely restricted matchings in interval graphs. SIAM J. Discrete Math. 32(1), 148–172 (2018)

    Article  MathSciNet  Google Scholar 

  4. Goddard, W., Hedetniemi, S.M., Hedetniemi, S.T., Laskar, R.: Generalized subgraph-restricted matchings in graphs. Discrete Math. 293(1), 129–138 (2005)

    Article  MathSciNet  Google Scholar 

  5. Goddard, W., Henning, M.A.: Independent domination in graphs: a survey and recent results. Discrete Math. 313(7), 839–854 (2013)

    Article  MathSciNet  Google Scholar 

  6. Golumbic, M.C., Hirst, T., Lewenstein, M.: Uniquely restricted matchings. Algorithmica 31(2), 139–154 (2001)

    Article  MathSciNet  Google Scholar 

  7. Jamison, R.E., Laskar, R.: Elimination orderings of chordal graphs. Combin. Appl. 192–200 (1982)

    Google Scholar 

  8. Kann, V.: On the approximability of NP-complete optimization problems. Ph.D. thesis, Royal Institute of Technology Stockholm (1992)

    Google Scholar 

  9. Lai, T.H., Wei, S.S.: Bipartite permutation graphs with application to the minimum buffer size problem. Discrete Appl. Math. 74(1), 33–55 (1997)

    Article  MathSciNet  Google Scholar 

  10. Mishra, S.: On the maximum uniquely restricted matching for bipartite graphs. Electron. Notes Discrete Math. 37, 345–350 (2011)

    Article  MathSciNet  Google Scholar 

  11. Müller, H.: Alternating cycle-free matchings. Order 7(1), 11–21 (1990)

    Article  MathSciNet  Google Scholar 

  12. Panda, B.S., Chaudhary, J.: Acyclic matching in some subclasses of graphs. In: Gąsieniec, L., Klasing, R., Radzik, T. (eds.) IWOCA 2020. LNCS, vol. 12126, pp. 409–421. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-48966-3_31

    Chapter  Google Scholar 

  13. Panda, B.S., Das, S.K.: A linear time recognition algorithm for proper interval graphs. Inf. Process. Lett. 87(3), 153–161 (2003)

    Article  MathSciNet  Google Scholar 

  14. Panda, B.S., Pandey, A.: On the complexity of minimum cardinality maximal uniquely restricted matching in graphs. In: Arumugam, S., Bagga, J., Beineke, L.W., Panda, B.S. (eds.) ICTCSDM 2016. LNCS, vol. 10398, pp. 218–227. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-64419-6_29

    Chapter  Google Scholar 

  15. Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science and Application Group, Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

    Juhi Chaudhary & B. S. Panda

Authors
  1. Juhi Chaudhary
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. S. Panda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. S. Panda .

Editor information

Editors and Affiliations

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

    Weili Wu

  2. Xiamen University, Xiamen, China

    Zhongnan Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chaudhary, J., Panda, B.S. (2020). On the Complexity of Minimum Maximal Uniquely Restricted Matching. In: Wu, W., Zhang, Z. (eds) Combinatorial Optimization and Applications. COCOA 2020. Lecture Notes in Computer Science(), vol 12577. Springer, Cham. https://doi.org/10.1007/978-3-030-64843-5_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-64843-5_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-64842-8

  • Online ISBN: 978-3-030-64843-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation