Skip to main content
SpringerLink
Log in

Long Circuits and Large Euler Subgraphs

  • Conference paper
Algorithms – ESA 2013 (ESA 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8125))

Included in the following conference series:

Abstract

An undirected graph is Eulerian if it is connected and all its vertices are of even degree. Similarly, a directed graph is Eulerian, if for each vertex its in-degree is equal to its out-degree. It is well known that Eulerian graphs can be recognized in polynomial time while the problems of finding a maximum Eulerian subgraph or a maximum induced Eulerian subgraph are NP-hard. In this paper, we study the parameterized complexity of the following Euler subgraph problems:

  • Large Euler Subgraph: For a given graph G and integer parameter k, does G contain an induced Eulerian subgraph with at least k vertices?

  • Long Circuit: For a given graph G and integer parameter k, does G contain an Eulerian subgraph with at least k edges?

Our main algorithmic result is that Large Euler Subgraph is fixed parameter tractable (FPT) on undirected graphs. We find this a bit surprising because the problem of finding an induced Eulerian subgraph with exactly k vertices is known to be W[1]-hard. The complexity of the problem changes drastically on directed graphs. On directed graphs we obtained the following complexity dichotomy: Large Euler Subgraph is NP-hard for every fixed k > 3 and is solvable in polynomial time for k ≤ 3. For Long Circuit, we prove that the problem is FPT on directed and undirected graphs.

Supported by the European Research Council (ERC) via grant Rigorous Theory of Preprocessing, reference 267959.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berman, P., Karpinski, M., Scott, A.D.: Approximation hardness of short symmetric instances of MAX-3SAT. Electronic Colloquium on Computational Complexity, ECCC (049) (2003)

    Google Scholar 

  3. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, L., Yang, B.: Parameterized complexity of even/odd subgraph problems. J. Discrete Algorithms 9(3), 231–240 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Courcelle, B.: The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues. ITA 26, 257–286 (1992)

    MathSciNet  MATH  Google Scholar 

  7. Cygan, M., Marx, D., Pilipczuk, M., Pilipczuk, M., Schlotter, I.: Parameterized complexity of Eulerian deletion problems. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 131–142. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  8. Dorn, F., Moser, H., Niedermeier, R., Weller, M.: Efficient algorithms for eulerian extension and rural postman. SIAM J. Discrete Math. 27(1), 75–94 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Downey, R.G., Fellows, M.R.: Parameterized complexity. Monographs in Computer Science. Springer, New York (1999)

    Book  Google Scholar 

  10. Erdös, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Math. 2, 463–470 (1935)

    MathSciNet  Google Scholar 

  11. Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410(1), 53–61 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fleischner, H.: Eulerian Graphs and Related Topics, Part 1, Amsterdam. Annals of Discrete Mathematics 45, vol. 1 (1990)

    Google Scholar 

  13. Flum, J., Grohe, M.: Parameterized complexity theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)

    Google Scholar 

  14. Fomin, F.V., Golovach, P.A.: Parameterized complexity of connected even/odd subgraph problems. In: STACS 2012. LIPIcs, vol. 14, pp. 432–440. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2012)

    Google Scholar 

  15. Gabow, H.N., Nie, S.: Finding a long directed cycle. ACM Transactions on Algorithms 4(1) (2008)

    Google Scholar 

  16. Garey, M.R., Johnson, D.S., Tarjan, R.E.: The planar hamiltonian circuit problem is NP-complete. SIAM J. Comput. 5(4), 704–714 (1976)

    Article  MathSciNet  Google Scholar 

  17. Golovach, P.A., Thilikos, D.M.: Paths of bounded length and their cuts: Parameterized complexity and algorithms. Discrete Optimization 8(1), 72–86 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kosowski, A., Li, B., Nisse, N., Suchan, K.: k-Chordal graphs: From cops and robber to compact routing via treewidth. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 610–622. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  19. Niedermeier, R.: Invitation to fixed-parameter algorithms, Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, PB 7803, 5020, Bergen, Norway

    Fedor V. Fomin & Petr A. Golovach

Authors
  1. Fedor V. Fomin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petr A. Golovach
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Utrecht University, Princetonplein 5, 3584 CC, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma “Tor Vergata”, via del Politecnico 1, 00133, Rome, Italy

    Giuseppe F. Italiano

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Fomin, F.V., Golovach, P.A. (2013). Long Circuits and Large Euler Subgraphs. In: Bodlaender, H.L., Italiano, G.F. (eds) Algorithms – ESA 2013. ESA 2013. Lecture Notes in Computer Science, vol 8125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40450-4_42

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-40450-4_42

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40449-8

  • Online ISBN: 978-3-642-40450-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation