Skip to main content
Springer Nature Link
Log in

Adjacencies for Structuring the Digital Plane

  • Conference paper
Combinatorial Image Analaysis (IWCIA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 7655))

Included in the following conference series:

  • 640 Accesses

Abstract

We study graphs with the vertex set ℤ2 which are subgraphs of the 8-adjacency graph and have the property that certain natural cycles in these graphs are Jordan curves, i.e., separate ℤ2 into exactly two connected components. For the minimal graphs with this property, we discuss their quotient graphs, too.

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. Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer (2008)

    Google Scholar 

  2. Khalimsky, E.D., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topology Appl. 36, 1–17 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Khalimsky, E.D., Kopperman, R., Meyer, P.R.: Boundaries in digital planes. Journ. Appl. Math. Stoch. Anal. 3, 27–55 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kiselman, C.O.: Digital Jordan Curve Theorems. In: Nyström, I., Sanniti di Baja, G., Borgefors, G. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 46–56. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  5. Kong, T.Y., Kopperman, R., Meyer, P.R.: A topological approach to digital topology. Amer. Math. Monthly 98, 902–917 (1991)

    Article  MathSciNet  Google Scholar 

  6. Kopperman, R., Meyer, P.R., Wilson, R.G.: A Jordan surface theorem for three-dimensional digital spaces. Discr. Comput. Geom. 6, 155–161 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Rosenfeld, A.: Digital topology. Amer. Math. Monthly 86, 621–630 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rosenfeld, A.: Picture Languages. Academic Press, New York (1979)

    MATH  Google Scholar 

  9. Šlapal, J.: Closure operations for digital topology. Theor. Comp. Sci. 305, 457–471 (2003)

    Article  MATH  Google Scholar 

  10. Šlapal, J.: A digital analogue of the Jordan curve theorem. Discr. Appl. Math. 139, 231–251 (2004)

    Article  MATH  Google Scholar 

  11. Šlapal, J.: Digital Jordan curves. Topology Appl. 153, 3255–3264 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Šlapal, J.: A quotient universal digital topology. Theor. Comp. Sci. 405, 164–175 (2008)

    Article  MATH  Google Scholar 

  13. Šlapal, J.: Jordan Curve Theorems with Respect to Certain Pretopologies on ℤ2. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 252–262. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  14. Šlapal, J.: Convenient Closure Operators on ℤ2. In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 425–436. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  15. Slapal, J.: A Jordan Curve Theorem in the Digital Plane. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds.) IWCIA 2011. LNCS, vol. 6636, pp. 120–131. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  16. Šlapal, J.: A digital pretopology and one of its quotients. Topology Proc. 39, 13–25 (2012)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Brno University of Technology, 616 69, Brno, Czech Republic

    Josef Šlapal

Authors
  1. Josef Šlapal
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer and Information Sciences, State University of New York at Fredonia, 280 Central Ave, 14063, Fredonia, NY, USA

    Reneta P. Barneva

  2. Mathematics Department, SUNY Buffalo State College, 1300Elmwood Ave, 14222, Buffalo, NY, USA

    Valentin E. Brimkov

  3. Department of Electrical and Computer Engineering, University of Texas at Austin, 1 University Station C0803, 78712-0240 TX, Austin, USA

    Jake K. Aggarwal

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Šlapal, J. (2012). Adjacencies for Structuring the Digital Plane. In: Barneva, R.P., Brimkov, V.E., Aggarwal, J.K. (eds) Combinatorial Image Analaysis. IWCIA 2012. Lecture Notes in Computer Science, vol 7655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34732-0_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-34732-0_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-34731-3

  • Online ISBN: 978-3-642-34732-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics