# KD-(k0, k1)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS

• Han, Sang-Eon (FACULTY OF LIBERAL EDUCATION INSTITUTE OF PURE AND APPLIED MATHEMATICS CHONBUK NATIONAL UNIVERSITY)
• Published : 2010.09.01
• 143 12

#### Abstract

Let $\mathbb{Z}^n$ be the Cartesian product of the set of integers $\mathbb{Z}$ and let ($\mathbb{Z}$, T) and ($\mathbb{Z}^n$, $T^n$) be the Khalimsky line topology on $\mathbb{Z}$ and the Khalimsky product topology on $\mathbb{Z}^n$, respectively. Then for a set $X\;{\subset}\;\mathbb{Z}^n$, consider the subspace (X, $T^n_X$) induced from ($\mathbb{Z}^n$, $T^n$). Considering a k-adjacency on (X, $T^n_X$), we call it a (computer topological) space with k-adjacency and use the notation (X, k, $T^n_X$) := $X_{n,k}$. In this paper we introduce the notions of KD-($k_0$, $k_1$)-homotopy equivalence and KD-k-deformation retract and investigate a classification of (computer topological) spaces $X_{n,k}$ in terms of a KD-($k_0$, $k_1$)-homotopy equivalence.

#### Keywords

computer topology;digital topology;digital space;KD-($k_0$, $k_1$)-continuity;KD-k-deformation retract;digital homotopy equivalence;KD-($k_0$, $k_1$)-homotopy equivalence;KD-k-homotopic thinning

#### Acknowledgement

Supported by : National Foundation of Korea(NRF)

#### References

1. P. Alexandroff, Diskrete Raume, Mat. Sb. 2 (1937), 501-519. https://doi.org/10.1070/SM1967v002n04ABEH002351
2. R. Ayala, E. Dominguez, A. R. Frances, and A. Quintero, Homotopy in digital spaces, Discrete Appl. Math. 125 (2003), no. 1, 3-24. https://doi.org/10.1016/S0166-218X(02)00221-4
3. L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833-839. https://doi.org/10.1016/0167-8655(94)90012-4
4. L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vision 10 (1999), no. 1, 51-62. https://doi.org/10.1023/A:1008370600456
5. L. Boxer, Properties of digital homotopy, J. Math. Imaging Vision 22 (2005), no. 1, 19-26. https://doi.org/10.1007/s10851-005-4780-y
6. J. Dontchev and Haruo Maki, Groups of ${\theta}$-generalized homeomorphisms and the digital line, Topology Appl. 95 (1999), no. 2, 113-128. https://doi.org/10.1016/S0166-8641(98)00004-2
7. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, A Compendium of Continuous Lattices, Springer-Verlag, Berlin-New York, 1980.
8. S. E. Han, On the classification of the digital images up to a digital homotopy equivalence, The Journal of Computer and Communications Research 10 (2000), 194-207.
9. S. E. Han, Computer topology and its applications, Honam Math. J. 25 (2003), no. 1, 153-162.
10. S. E. Han, Comparison between digital continuity and computer continuity, Honam Math. J. 26 (2004), no. 3, 331-339.
11. S. E. Han, Minimal digital pseudotorus with k-adjacency, k ${\in}$ 6, 18, 26, Honam Math. J. 26 (2004), no. 2, 237-246.
12. S. E. Han, Algorithm for discriminating digital images with respect to a digital $(k_{0},\;k_{1})$-homeomorphism, Jour. of Applied Mathematics and Computing 18 (2005), no. 1-2, 505-512.
13. S. E. Han, Non-product property of the digital fundamental group, Inform. Sci. 171 (2005), no. 1-3, 73-91. https://doi.org/10.1016/j.ins.2004.03.018
14. S. E. Han, On the simplicial complex stemmed from a digital graph, Honam Math. J. 27 (2005), no. 1, 115-129.
15. S. E. Han, Connected sum of digital closed surfaces, Inform. Sci. 176 (2006), no. 3, 332-348. https://doi.org/10.1016/j.ins.2004.11.003
16. S. E. Han, Discrete Homotopy of a Closed k-Surface, LNCS 4040, Springer-Verlag, Berlin, pp. 214-225 (2006).
17. S. E. Han, Erratum to: “Non-product property of the digital fundamental group”, Inform. Sci. 176 (2006), no. 2, 215-216. https://doi.org/10.1016/j.ins.2005.03.014
18. S. E. Han, Remarks on digital homotopy equivalence, Honam Math. J. 29 (2007), no. 1, 101-118. https://doi.org/10.5831/HMJ.2007.29.1.101
19. S. E. Han, Strong k-deformation retract and its applications, J. Korean Math. Soc. 44 (2007), no. 6, 1479-1503. https://doi.org/10.4134/JKMS.2007.44.6.1479
20. S. E. Han, Continuities and homeomorphisms in computer topology and their applications, J. Korean Math. Soc. 45 (2008), no. 4, 923-952. https://doi.org/10.4134/JKMS.2008.45.4.923
21. S. E. Han, Equivalent $(k_{0},\;k_{1})$-covering and generalized digital lifting, Inform. Sci. 178 (2008), no. 2, 550-561. https://doi.org/10.1016/j.ins.2007.02.004
22. S. E. Han, The k-homotopic thinning and a torus-like digital image in $Z^n$, J. Math. Imaging Vision 31 (2008), no. 1, 1-16. https://doi.org/10.1007/s10851-007-0061-2
23. S. E. Han, Map preserving local properties of a digital image, Acta Appl. Math. 104 (2008), no. 2, 177-190. https://doi.org/10.1007/s10440-008-9250-2
24. S. E. Han, Extension of several continuities in computer topology, Bull. Korean Math. Soc., submitted.
25. S. E. Han and N. D. Georgiou, On computer topological function space, J. Korean Math. Soc. 46 (2009), no. 4, 841-857. https://doi.org/10.4134/JKMS.2009.46.4.841
26. 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).
27. E. Khalimsky, R. Kopperman, and P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology and its Applications 36 (1991), no. 1, 1-17. https://doi.org/10.1016/0166-8641(90)90031-V
28. I.-S. Kim and S. E. Han, Digital covering theory and its applications, Honam Math. J. 30 (2008), no. 4, 589-602. https://doi.org/10.5831/HMJ.2008.30.4.589
29. I.-S. Kim, S. E. Han, and C. J. Yoo, The almost pasting property of digital continuity, Acta Applicandae Mathematicae 110 (1) (2010), 399-408. https://doi.org/10.1007/s10440-008-9422-0
30. T. Y. Kong, A digital fundamental group, Computers and Graphics 13 (1989), 159-166. https://doi.org/10.1016/0097-8493(89)90058-7
31. T. Y. Kong and A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996.
32. R. Malgouyres, Homotopy in two-dimensional digital images, Theoret. Comput. Sci. 230 (2000), no. 1-2, 221-233. https://doi.org/10.1016/S0304-3975(98)00347-8
33. E. Melin, Extension of continuous functions in digital spaces with the Khalimsky topology, Topology Appl. 153 (2005), no. 1, 52-65. https://doi.org/10.1016/j.topol.2004.12.004
34. A. Rosenfeld, Digital topology, Amer. Math. Monthly 86 (1979), no. 8, 621-630. https://doi.org/10.2307/2321290
35. J. Slapal, Digital Jordan curves, Topology Appl. 153 (2006), no. 17, 3255-3264. https://doi.org/10.1016/j.topol.2005.10.011

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