Skip to main content
SpringerLink
Log in

Disconnected Matchings

  • Conference paper
  • First Online:
Computing and Combinatorics (COCOON 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13025))

Included in the following conference series:

Abstract

In 2005, Goddard, Hedetniemi, Hedetniemi and Laskar [Generalized subgraph-restricted matchings in graphs, Discrete Mathematics, 293 (2005) 129 – 138] asked the computational complexity of determining the maximum cardinality of a matching whose vertex set induces a disconnected graph. In this paper we answer this question. In fact, we consider the generalized problem of finding c-disconnected matchings; such matchings are ones whose vertex sets induce subgraphs with at least c connected components. We show that, for every fixed \(c \ge 2\), this problem is \(\mathsf {NP}\)-\(\mathsf {complete}\) even if we restrict the input to bounded diameter bipartite graphs. For the case when c is part of the input, we show that the problem is \(\mathsf {NP}\)-\(\mathsf {complete}\) for chordal graphs while being solvable in polynomial time for interval graphs, \(\mathsf {FPT}\) when parameterized by treewidth, and \(\mathsf {XP}\) for graphs with a polynomial number of minimal separators, when parameterized by c.

G. C. M. Gomes—Partially supported by CAPES

V. F. dos Santos—Partially supported by FAPEMIG and CNPq

J. L. Szwarcfiter—Partially supported by CNPq.

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

Notes

  1. 1.

    A line-complete matching M is a matching such that every pair of edges of M has a common adjacent edge.

References

  1. Abrishami, T., Chudnovsky, M., Dibek, C., Thomassé, S., Trotignon, N., Vušković, K.: Graphs with polynomially many minimal separators. arXiv preprint arXiv:2005.05042 (2020)

  2. Baste, J., Rautenbach, D.: Degenerate matchings and edge colorings. Discret. Appl. Math. 239, 38–44 (2018). https://doi.org/10.1016/j.dam.2018.01.002

    Article  MathSciNet  MATH  Google Scholar 

  3. Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Heidelberg (2008). https://www.springer.com/gp/book/9781846289699

  4. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1999)

    Google Scholar 

  5. Cameron, K.: Induced matchings. Discret. Appl. Math. 24(1), 97–102 (1989). https://doi.org/10.1016/0166-218X(92)90275-F

    Article  MathSciNet  MATH  Google Scholar 

  6. Cameron, K.: Connected matchings. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization — Eureka, You Shrink! LNCS, vol. 2570, pp. 34–38. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36478-1_5

    Chapter  Google Scholar 

  7. Chandran, L.S.: A linear time algorithm for enumerating all the minimum and minimal separators of a chordal graph. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, pp. 308–317. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44679-6_34

    Chapter  MATH  Google Scholar 

  8. Cygan, M., et al.: Parameterized Algorithms, vol. 3. Springer, Heidelberg (2015)

    Book  Google Scholar 

  9. Deogun, J.S., Kloks, T., Kratsch, D., Müller, H.: On the vertex ranking problem for trapezoid, circular-arc and other graphs. Discret. Appl. Math. 98(1), 39–63 (1999). https://doi.org/10.1016/S0166-218X(99)00179-1

    Article  MathSciNet  MATH  Google Scholar 

  10. Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965). https://doi.org/10.4153/CJM-1965-045-4

    Article  MathSciNet  MATH  Google Scholar 

  11. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  12. Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Can. J. Math. 16, 539–548 (1964). https://doi.org/10.4153/CJM-1964-055-5

    Article  MathSciNet  MATH  Google Scholar 

  13. Goddard, W., Hedetniemi, S.M., Hedetniemi, S.T., Laskar, R.: Generalized subgraph-restricted matchings in graphs. Discret. Math. 293(1), 129–138 (2005). https://doi.org/10.1016/j.disc.2004.08.027

    Article  MathSciNet  MATH  Google Scholar 

  14. Golumbic, M.C., Hirst, T., Lewenstein, M.: Uniquely restricted matchings. Algorithmica 31(2), 139–154 (2001). https://doi.org/10.1007/s00453-001-0004-z

    Article  MathSciNet  MATH  Google Scholar 

  15. Kobler, D., Rotics, U.: Finding maximum induced matchings in subclasses of claw-free and p5-free graphs, and in graphs with matching and induced matching of equal maximum size. Algorithmica 37(4), 327–346 (2003). https://doi.org/10.1007/s00453-003-1035-4

    Article  MathSciNet  MATH  Google Scholar 

  16. Lozin, V.V.: On maximum induced matchings in bipartite graphs. Inf. Process. Lett. 81(1), 7–11 (2002). https://doi.org/10.1016/S0020-0190(01)00185-5

    Article  MathSciNet  MATH  Google Scholar 

  17. Masquio, B.P.: Emparelhamentos desconexos. Master’s thesis, Universidade do Estado do Rio de Janeiro (2019). http://www.bdtd.uerj.br/handle/1/7663

  18. Micali, S., Vazirani, V.V.: An \({O}(\sqrt{|V|}|{E}|)\) algorithm for finding maximum matching in general graphs. In: 21st Annual Symposium on Foundations of Computer Science, pp. 17–27, October 1980. https://doi.org/10.1109/SFCS.1980.12

  19. Moser, H., Sikdar, S.: The parameterized complexity of the induced matching problem. Discret. Appl. Math. 157(4), 715–727 (2009). https://doi.org/10.1016/j.dam.2008.07.011

    Article  MathSciNet  MATH  Google Scholar 

  20. 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 

  21. Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986). https://doi.org/10.1016/0196-6774(86)90023-4

  22. Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing. STOC 1978, pp. 216–226. Association for Computing Machinery, New York (1978). https://doi.org/10.1145/800133.804350

  23. Shen, H., Liang, W.: Efficient enumeration of all minimal separators in a graph. Theoret. Comput. Sci. 180(1–2), 169–180 (1997)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Ciência da Computação, Universidade Federal de Minas Gerais (UFMG), Belo Horizonte, MG, Brazil

    Guilherme C. M. Gomes & Vinicius F. dos Santos

  2. Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro (UERJ), Rio de Janeiro, RJ, Brazil

    Bruno P. Masquio, Paulo E. D. Pinto & Jayme L. Szwarcfiter

  3. Instituto de Matemática e PESC/COPPE, Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, RJ, Brazil

    Jayme L. Szwarcfiter

Authors
  1. Guilherme C. M. Gomes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bruno P. Masquio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paulo E. D. Pinto
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vinicius F. dos Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jayme L. Szwarcfiter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bruno P. Masquio .

Editor information

Editors and Affiliations

  1. National Cheng Kung University, Tainan, Taiwan

    Chi-Yeh Chen

  2. National Tsing Hua University, Hsinchu, Taiwan

    Wing-Kai Hon

  3. National Taipei University of Business, Taoyuan, Taiwan

    Ling-Ju Hung

  4. National Taitung University, Taitung, Taiwan

    Chia-Wei Lee

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gomes, G.C.M., Masquio, B.P., Pinto, P.E.D., dos Santos, V.F., Szwarcfiter, J.L. (2021). Disconnected Matchings. In: Chen, CY., Hon, WK., Hung, LJ., Lee, CW. (eds) Computing and Combinatorics. COCOON 2021. Lecture Notes in Computer Science(), vol 13025. Springer, Cham. https://doi.org/10.1007/978-3-030-89543-3_48

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-89543-3_48

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-89542-6

  • Online ISBN: 978-3-030-89543-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation