Structuring Digital Spaces by Path-Partition Induced Closure Operators on Graphs

  • Josef ŠlapalEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10149)


We study closure operators on graphs which are induced by path partitions, i.e., certain sets of paths of the same lengths in these graphs. We investigate connectedness with respect to the closure operators studied. In particular, the closure operators are discussed that are induced by path partitions of some natural graphs on the digital spaces \({\mathbb {Z}}^n\), \(n>0\) a natural number. For the case \(n=2\), i.e., for the digital plane \({\mathbb {Z}}^2\), the induced closure operators are shown to satisfy an analogue of the Jordan curve theorem, which allows using them as convenient background structures for studying digital images.


Graphs Closure operators Path-partition Connectedness Jordan curve 



This work was supported by Ministry of Education, Youth and Sports of the Czech Republic from the National Programme of Sustainability (NPU II) project “IT4Innovations Excellence in Science - LQ1602”.


  1. 1.
    Čech, E.: Topological spaces. In: Topological Papers of Eduard Čech, pp. 436–472. Academia, Prague (1968)Google Scholar
  2. 2.
    Čech, E.: Topological Spaces. Academia, Prague (1966). (revised by Z. Frolík and M. Katětov)zbMATHGoogle Scholar
  3. 3.
    Engelking, R.: General Topology. Państwowe Wydawnictwo Naukowe, Warszawa (1977)zbMATHGoogle Scholar
  4. 4.
    Harrary, F.: Graph Theory. Addison-Wesley Publ. Comp., Reading (1969)Google Scholar
  5. 5.
    Khalimsky, E.D., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topol. Appl. 36, 1–17 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kong, T.Y., Roscoe, W.: A theory of binary digital pictures. Comput. Vis. Graph. Image Process. 32, 221–243 (1985)CrossRefzbMATHGoogle Scholar
  7. 7.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48, 357–393 (1989)CrossRefGoogle Scholar
  8. 8.
    Kong, T.Y., Kopperman, R., Meyer, P.R.: A topological approach to digital topology. Am. Math. Mon. 98, 902–917 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rosenfeld, A.: Connectivity in digital pictures. J. Assoc. Comput. Mach. 17, 146–160 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rosenfeld, A.: Digital topology. Am. Math. Mon. 86, 621–630 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Šlapal, J.: Direct arithmetics of relational systems. Publ. Math. Debr. 38, 39–48 (1991)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Šlapal, J.: A digital analogue of the Jordan curve theorem. Discret. Appl. Math. 139, 231–251 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Šlapal, J.: Convenient closure operators on \(\mathbb{Z}^2\). In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 425–436. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-10210-3_33 CrossRefGoogle Scholar
  14. 14.
    Šlapal, J.: A quotient universal digital topology. Theor. Comput. Sci. 405, 164–175 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Šlapal, J.: Graphs with a walk partition for structuring digital spaces. Inf. Sci. 233, 305–312 (2013)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.IT4Innovations Centre of ExcellenceBrno University of TechnologyBrnoCzech Republic

Personalised recommendations