호남수학학술지 (Honam Mathematical Journal)

  • 제37권1호
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  • Pages.135-147
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  • 2015
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  • 1225-293X(pISSN)
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  • 2288-6176(eISSN)
호남수학회 (The Honam Mathematical Society)
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COMPARISON AMONG SEVERAL ADJACENCY PROPERTIES FOR A DIGITAL PRODUCT

  • Han, Sang-Eon (Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University)
  • 투고 : 2014.11.16
  • 심사 : 2015.02.27
  • 발행 : 2015.03.25
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초록

Owing to the notion of a normal adjacency for a digital product in [8], the study of product properties of digital topological properties has been substantially done. To explain a normal adjacency of a digital product more efficiently, the recent paper [22] proposed an S-compatible adjacency of a digital product. Using an S-compatible adjacency of a digital product, we also study product properties of digital topological properties, which improves the presentations of a normal adjacency of a digital product in [8]. Besides, the paper [16] studied the product property of two digital covering maps in terms of the $L_S$- and the $L_C$-property of a digital product which plays an important role in studying digital covering and digital homotopy theory. Further, by using HS- and HC-properties of digital products, the paper [18] studied multiplicative properties of a digital fundamental group. The present paper compares among several kinds of adjacency relations for digital products and proposes their own merits and further, deals with the problem: consider a Cartesian product of two simple closed $k_i$-curves with $l_i$ elements in $Z^{n_i}$, $i{\in}\{1,2\}$ denoted by $SC^{n_1,l_1}_{k_1}{\times}SC^{n_2,l_2}_{k_2}$. Since a normal adjacency for this product and the $L_C$-property are different from each other, the present paper address the problem: for the digital product does it have both a normal k-adjacency of $Z^{n_1+n_2}$ and another adjacency satisfying the $L_C$-property? This research plays an important role in studying product properties of digital topological properties.

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참고문헌

  1. C. Berge, Graphs and Hypergraphs, 2nd ed., North-Holland, Amsterdam, 1976.
  2. G. Bertrand, Simple points, topological numbers and geodesic neighborhoods in cubic grids, Pattern Recognition Letters 15 (1994), 1003-1011. https://doi.org/10.1016/0167-8655(94)90032-9
  3. G. Bertrand and R. Malgouyres, Some topological properties of discrete surfaces, Jour. of Mathematical Imaging and Vision 20 (1999) 207-221.
  4. 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
  5. L. Chen, Discrete Surfaces and Manifolds: A Theory of Digital Discrete Geometry and Topology, Scientific and Practical Computing, Rockville, 2004.
  6. S. E. Han, On the classification of the digital images up to a digital homotopy equivalence, The Jour. of Computer and Communications Research 10 (2000), 194-207.
  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, Discrete Homotopy of a Closed k-Surface LNCS 4040, Springer-Verlag, Berlin, pp.214-225 (2006).
  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, Minimal simple closed 18-surfaces and a topological preservation of 3D surfaces, Information Sciences 176(2)(2006) 120-134. https://doi.org/10.1016/j.ins.2005.01.002
  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, 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
  15. 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
  16. S.E. Han, Cartesian product of the universal covering property, Acta Applicandae Mathematicae 108 (2010) 363-383.
  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, Multiplicative property of the digital fundamental group, Acta Applicandae Mathematicae 110(2) (2010) 921-944. https://doi.org/10.1007/s10440-009-9486-5
  19. S.E. Han, Ultra regular covering space and its automorphism group International Journal of Applied Mathematics & Computer Science, (2010) 20(4) 699-710. https://doi.org/10.2478/v10006-010-0053-z
  20. S.E. Han, Non-ultra regular digital covering spaces with nontrivial automorphism groups Filomat, (2013) 27(7) 1205-1218. https://doi.org/10.2298/FIL1307205H
  21. S.E. Han, Study on topological spaces with the semi-$T_{\frac{1}{2}}$ separation axiom, Honam Mathematical Journal (2013) 35(4) 707-716. https://doi.org/10.5831/HMJ.2013.35.4.707
  22. S.E. Han, An equivalent property of a normal adjacency of a digital product, Honam Mathematical Journal (2014) 36(1) 199-215. https://doi.org/10.5831/HMJ.2014.36.1.199
  23. S.E. Han, Remarks on simply k-connectivity and k-deformation retract in digital topology, Honam Mathematical Journal (2014) 36(3) 519-530. https://doi.org/10.5831/HMJ.2014.36.3.519
  24. S.E. Han and Sik Lee, Remarks on digital products with normal adjacency relations, Honam Mathematical Journal 35 (3)(2013) 515-524. https://doi.org/10.5831/HMJ.2013.35.3.515
  25. S.E. Han and Sik Lee, Utility of digital covering theory, Honam Mathematical Journal (2014) 36(3) 695-706. https://doi.org/10.5831/HMJ.2014.36.3.695
  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. F. Harary, Graph theory, Addison-Wesley Publishing, Reading, MA, 1969.
  28. G. T. Herman, Geometry of Digital Spaces Birkhauser, Boston, 1998.
  29. T.Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, (1996).
  30. 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).
  31. A. Rosenfeld, Digital topology, Amer. Math. Mon. 86 (1979) 76-87.