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REMARKS ON SIMPLY k-CONNECTIVITY AND k-DEFORMATION RETRACT IN DIGITAL TOPOLOGY
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  • Journal title : Honam Mathematical Journal
  • Volume 36, Issue 3,  2014, pp.519-530
  • Publisher : The Honam Mathematical Society
  • DOI : 10.5831/HMJ.2014.36.3.519
 Title & Authors
REMARKS ON SIMPLY k-CONNECTIVITY AND k-DEFORMATION RETRACT IN DIGITAL TOPOLOGY
Han, Sang-Eon;
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 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
simply k-connected;digital isomorphism;strong k-deformation;weak k-deformation retract;k-homotopy equivalence (same k-homotopy type);digital topology;
 Language
English
 Cited by
1.
UTILITY OF DIGITAL COVERING THEORY,;;

호남수학학술지, 2014. vol.36. 3, pp.695-706 crossref(new window)
2.
COMPARISON AMONG SEVERAL ADJACENCY PROPERTIES FOR A DIGITAL PRODUCT,;

호남수학학술지, 2015. vol.37. 1, pp.135-147 crossref(new window)
1.
COMPARISON AMONG SEVERAL ADJACENCY PROPERTIES FOR A DIGITAL PRODUCT, Honam Mathematical Journal, 2015, 37, 1, 135  crossref(new windwow)
2.
UTILITY OF DIGITAL COVERING THEORY, Honam Mathematical Journal, 2014, 36, 3, 695  crossref(new windwow)
 References
1.
L. Boxer, Digitally continuous functions, Pattern Recognition Letters, 15 (1994), 833-839. crossref(new window)

2.
L. Boxer, A classical construction for the digital fundamental group, Jour. of Mathematical Imaging and Vision 10 (1999), 51-62. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

14.
S.E. Han, Comparison among digital k-fundamental groups and its applications, Information Sciences 178 (2008), 2091-2104. crossref(new window)

15.
S.E. Han, Equivalent ($k_0$, $k_1$)-covering and generalized digital lifting, Information Sciences 178(2) (2008), 550-561. crossref(new window)

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. crossref(new window)

17.
S.E. Han, KD-($k_0$, $k_1$)-homotopy equivalence and its applications, Journal of Korean Mathematical Society 47(5) (2010), 1031-1054. crossref(new window)

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. crossref(new window)

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.