Browse > Article
http://dx.doi.org/10.4134/JKMS.2010.47.5.1031

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

Han, Sang-Eon (FACULTY OF LIBERAL EDUCATION INSTITUTE OF PURE AND APPLIED MATHEMATICS CHONBUK NATIONAL UNIVERSITY)
Publication Information
Journal of the Korean Mathematical Society / v.47, no.5, 2010 , pp. 1031-1054 More about this Journal
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;
Citations & Related Records
Times Cited By KSCI : 5  (Citation Analysis)
Times Cited By Web Of Science : 3  (Related Records In Web of Science)
Times Cited By SCOPUS : 3
연도 인용수 순위
1 J. Slapal, Digital Jordan curves, Topology Appl. 153 (2006), no. 17, 3255-3264.   DOI   ScienceOn
2 S. E. Han, Map preserving local properties of a digital image, Acta Appl. Math. 104 (2008), no. 2, 177-190.   DOI
3 S. E. Han, Extension of several continuities in computer topology, Bull. Korean Math. Soc., submitted.
4 S. E. Han and N. D. Georgiou, On computer topological function space, J. Korean Math. Soc. 46 (2009), no. 4, 841-857.   과학기술학회마을   DOI   ScienceOn
5 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).
6 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.   DOI   ScienceOn
7 I.-S. Kim and S. E. Han, Digital covering theory and its applications, Honam Math. J. 30 (2008), no. 4, 589-602.   DOI   ScienceOn
8 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.   DOI   ScienceOn
9 T. Y. Kong, A digital fundamental group, Computers and Graphics 13 (1989), 159-166.   DOI   ScienceOn
10 T. Y. Kong and A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996.
11 R. Malgouyres, Homotopy in two-dimensional digital images, Theoret. Comput. Sci. 230 (2000), no. 1-2, 221-233.   DOI   ScienceOn
12 E. Melin, Extension of continuous functions in digital spaces with the Khalimsky topology, Topology Appl. 153 (2005), no. 1, 52-65.   DOI   ScienceOn
13 A. Rosenfeld, Digital topology, Amer. Math. Monthly 86 (1979), no. 8, 621-630.   DOI   ScienceOn
14 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.
15 S. E. Han, Non-product property of the digital fundamental group, Inform. Sci. 171 (2005), no. 1-3, 73-91.   DOI   ScienceOn
16 S. E. Han, On the simplicial complex stemmed from a digital graph, Honam Math. J. 27 (2005), no. 1, 115-129.
17 S. E. Han, Connected sum of digital closed surfaces, Inform. Sci. 176 (2006), no. 3, 332-348.   DOI   ScienceOn
18 S. E. Han, Discrete Homotopy of a Closed k-Surface, LNCS 4040, Springer-Verlag, Berlin, pp. 214-225 (2006).
19 S. E. Han, Erratum to: “Non-product property of the digital fundamental group”, Inform. Sci. 176 (2006), no. 2, 215-216.   DOI   ScienceOn
20 S. E. Han, Remarks on digital homotopy equivalence, Honam Math. J. 29 (2007), no. 1, 101-118.   DOI   ScienceOn
21 S. E. Han, Strong k-deformation retract and its applications, J. Korean Math. Soc. 44 (2007), no. 6, 1479-1503.   과학기술학회마을   DOI   ScienceOn
22 S. E. Han, Continuities and homeomorphisms in computer topology and their applications, J. Korean Math. Soc. 45 (2008), no. 4, 923-952.   과학기술학회마을   DOI   ScienceOn
23 R. Ayala, E. Dominguez, A. R. Frances, and A. Quintero, Homotopy in digital spaces, Discrete Appl. Math. 125 (2003), no. 1, 3-24.   DOI   ScienceOn
24 S. E. Han, Equivalent $(k_{0},\;k_{1})$-covering and generalized digital lifting, Inform. Sci. 178 (2008), no. 2, 550-561.   DOI   ScienceOn
25 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.   DOI
26 P. Alexandroff, Diskrete Raume, Mat. Sb. 2 (1937), 501-519.   DOI   ScienceOn
27 L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833-839.   DOI   ScienceOn
28 L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vision 10 (1999), no. 1, 51-62.   DOI
29 L. Boxer, Properties of digital homotopy, J. Math. Imaging Vision 22 (2005), no. 1, 19-26.   DOI
30 J. Dontchev and Haruo Maki, Groups of ${\theta}$-generalized homeomorphisms and the digital line, Topology Appl. 95 (1999), no. 2, 113-128.   DOI   ScienceOn
31 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.
32 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.
33 S. E. Han, Minimal digital pseudotorus with k-adjacency, k ${\in}$ 6, 18, 26, Honam Math. J. 26 (2004), no. 2, 237-246.
34 S. E. Han, Computer topology and its applications, Honam Math. J. 25 (2003), no. 1, 153-162.
35 S. E. Han, Comparison between digital continuity and computer continuity, Honam Math. J. 26 (2004), no. 3, 331-339.