Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

REMARKS ON SIMPLY k-CONNECTIVITY AND k-DEFORMATION RETRACT IN DIGITAL TOPOLOGY

  • Han, Sang-Eon (Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University)
  • Received : 2014.05.15
  • Accepted : 2014.06.10
  • Published : 2014.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

To study a deformation of a digital space from the viewpoint of digital homotopy theory, we have often used the notions of a weak k-deformation retract [20] and a strong k-deformation retract [10, 12, 13]. Thus the papers [10, 12, 13, 16] firstly developed the notion of a strong k-deformation retract which can play an important role in studying a homotopic thinning of a digital space. Besides, the paper [3] deals with a k-deformation retract and its homotopic property related to a digital fundamental group. Thus, as a survey article, comparing among a k-deformation retract in [3], a strong k-deformation retract in [10, 12, 13], a weak deformation k-retract in [20] and a digital k-homotopy equivalence [5, 24], we observe some relationships among them from the viewpoint of digital homotopy theory. Furthermore, the present paper deals with some parts of the preprint [10] which were not published in a journal (see Proposition 3.1). Finally, the present paper corrects Boxer's paper [3] as follows: even though the paper [3] referred to the notion of a digital homotopy equivalence (or a same k-homotopy type) which is a special kind of a k-deformation retract, we need to point out that the notion was already developed in [5] instead of [3] and further corrects the proof of Theorem 4.5 of Boxer's paper [3] (see the proof of Theorem 4.1 in the present paper). While the paper [4] refers some properties of a deck transformation group (or an automorphism group) of digital covering space without any citation, the study was early done by Han in his paper (see the paper [14]).

Keywords

References

  1. L. Boxer, Digitally continuous functions, Pattern Recognition Letters, 15 (1994), 833-839. https://doi.org/10.1016/0167-8655(94)90012-4
  2. L. Boxer, A classical construction for the digital fundamental group, Jour. of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456
  3. L. Boxer, Properties of digital homotopy, Jour. of Mathematical Imaging and Vision 10 (2005), 19-26.
  4. L. Boxer and I. Karaca, Some Properties of digital covering spaces, Jour. of Mathematical Imaging and Vision 37 (2010), 17-26. https://doi.org/10.1007/s10851-010-0189-3
  5. S.E. Han, On the classification of the digital images up to digital homotopy equivalence, Jour. Comput. Commun. Res. 10 (2000), 207-216.
  6. S.E. Han, Generalized digital ($k_0$, $k_1$)-homeomorphism, Note di Mathematica 22(2) (2003), 157-166.
  7. S.E. Han, Digital coverings and their applications, Jour. of Applied Mathematics and Computing 18(1-2) (2005), 487-495.
  8. S.E. Han, Non-product property of the digital fundamental group, Information Sciences 171(1-3) (2005), 73-91. https://doi.org/10.1016/j.ins.2004.03.018
  9. S.E. Han, On the simplicial complex stemmed from a digital graph, Honam Mathematical Journal 27(1) (2005), 115-129.
  10. S.E. Han, The k-fundamental group of a computer topological product space, Preprint submitted to Elsevier (2005), 1-23.
  11. S.E. Han, Erratum to "Non-product property of the digital fundamental group", Information Sciences 176(1) (2006), 215-216. https://doi.org/10.1016/j.ins.2005.03.014
  12. S.E. Han, Discrete Homotopy of a Closed k-Surface, LNCS 4040, Springer-Verlag, Berlin, (2006), 214-225.
  13. S.E. Han, Strong k-deformation retract and its applications, Journal of the Korean Mathematical Society 44(6) (2007), 1479-1503. https://doi.org/10.4134/JKMS.2007.44.6.1479
  14. S.E. Han, Comparison among digital k-fundamental groups and its applications, Information Sciences 178 (2008), 2091-2104. https://doi.org/10.1016/j.ins.2007.11.030
  15. S.E. Han, Equivalent ($k_0$, $k_1$)-covering and generalized digital lifting, Information Sciences 178(2) (2008), 550-561. https://doi.org/10.1016/j.ins.2007.02.004
  16. S.E. Han, The k-homotopic thinning and a torus-like digital image in $Z^n$, Journal of Mathematical Imaging and Vision 31(1) (2008), 1-16. https://doi.org/10.1007/s10851-007-0061-2
  17. S.E. Han, KD-($k_0$, $k_1$)-homotopy equivalence and its applications, Journal of Korean Mathematical Society 47(5) (2010), 1031-1054. https://doi.org/10.4134/JKMS.2010.47.5.1031
  18. S.E. Han, Ultra regular covering space and its automorphism group, International Journal of Applied Mathematics & Computer Science, 20(4) (2010), 699-710.
  19. S.E. Han, Non-ultra regular digital covering spaces with nontrivial automorphism groups, Filomat, 27(7) (2013), 1205-1218. https://doi.org/10.2298/FIL1307205H
  20. S.E. Han, Sik Lee, Weak k-deformation retract and its applications, Far East Journal of Mathematical Sciences, 30(1) (2008), 157-174.
  21. E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conferences on Systems, Man, and Cybernetics (1987), 227-234.
  22. T.Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996.
  23. W.S. Massey, Algebraic Topology, Springer-Verlag, New York, 1977.
  24. S.E. Han and B.G. Park, Digital graph ($k_0$, $k_1$)-homotopy equivalence and its applications, http://atlas-conferences.com/c/a/k/b/35.htm (2003).
  25. B.G. Park and S.E. Han, Classification of of digital graphs vian a digital graph ($k_0$, $k_1$)-isomorphism, http://atlas-conferences.com/c/a/k/b/36.htm (2003).
  26. A. Rosenfeld, Digital topology, Am. Math. Mon. 86 (1979), 76-87.

Cited by

  1. COMPARISON AMONG SEVERAL ADJACENCY PROPERTIES FOR A DIGITAL PRODUCT vol.37, pp.1, 2015, https://doi.org/10.5831/HMJ.2015.37.1.135
  2. UTILITY OF DIGITAL COVERING THEORY vol.36, pp.3, 2014, https://doi.org/10.5831/HMJ.2014.36.3.695