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[object Object],On Admissible Labelings and  Digital Topologies ,[object Object],Research partially funded by the  UP Natural Sciences Research Institute, Diliman, Quezon City
Digital Topology ,[object Object],[object Object],[object Object]
Z 2  and  Z 3
Neighbors Let  x  =  (n 1 ,n 2 ,...,n d  )  and  y = (m 1 ,m 2 ,...,m d  )   Z  d ie. ie. ,[object Object],[object Object]
2d-Connected M  ⊆   Z  d  2d-connected   if    x,y   ∈   M,   ∃   2d-path  from  x  to  y  in  M * analogous for  (3 d -1)-connected
Digital Topologies on  Z d D1 D2 D3 Topologies on Z d  satisfying
Counting Digital Topologies Khalimsky line: Marcus-Wyse Alexandroff-Hopf ,[object Object],[object Object]
Counting Digital Topologies ,[object Object],[object Object],[object Object]
Open Problem
Stable Sets of  Z 2 d M  ⊆  Z 2 d   where none of its  elements are  2d-neighbors Also called  stable subsets  of the hypercube  Q d (vertices are pairwise nonadjacent under  Q d ) S  := collection of stable sets of  Z 2 d
[object Object],[object Object],[object Object],[object Object],Orbits of Stable Sets
Admissible Labelings of  Z 2 d (3)  means no two points in  dom   are  2d-neighbors (3)  and  (4)  imply that    (x)   ≤  d-2      x  ∈   dom 
Admissible Labelings of  Z 2 d  1 ,   2   isomorphic   if  ∃   isometry   Φ   of  Z 2 d  onto  Z 2 d   s.t.  dom  1  =  Φ( dom  2 )   and    1  =   2  (  Φ ( dom  1 ) )   (0,1)-admissible labeling  if   (x)   ∈  { 0,1 }  otherwise,  non-(0,1)-admissible labeling
Correspondence isomorphism classes of admissible labelings  of  Z 2 d   homeomorphism classes of digital topologies on  Z d (Kong: using normalized admissible functions and digital strict partial orders) 1-1 correspondence
[object Object],[object Object],Kong's Algorithm
Counting Admissible Labelings We develop an algorithm to find and  count all isomorphism classes of  admissible labelings on  Z 2 d   ,[object Object],[object Object]
Results for  Z 5 Lower bound for number of digital topologies for d=5
Further Work ,[object Object],[object Object],[object Object],[object Object]
References U. Eckhart and L. Latecki . Topologies for the Digital Spaces Z2 and Z3. Computer Vision and Image Understanding 90 (2003) 295-312. R. Klette and A. Rosenfeld . Digital Geometry. Morgan Kauffman Publishers. San Francisco, CA. 2004. T.Y. Kong . The Khalimsky Topologies are precisely those simply connected topologies on Zn whose connected sets include all 2n-connected sets but no (3n-1)-disconnected sets. Theoretical Computer Science 305 (2003) 221-235. T.Y. Kong . Topological adjacency relations on Zn. Theoretical Computer Science 283 (2002) 3-28. T.Y. Kong, R. Kopperman and P.R. Meyer . A Topological Approach to Digital Topology. Amer. Math. Monthly 98 (1991), 901-917. R. Kopperman . The Khalimsky Line as a Foundation for Digital Topology. Shape in Picture (Proceedings of the NATO Advanced Research Workshop held at Driebergen, Spet. 1992). Springer. 1994 V. Kovalevsk y. Axiomatische Digitaltopologie. http://www.bv.inf.tu-dresden.de/AKTUELL/S_KOLL_0406/VORTRAEGE/V1_Kovalevski.ppt. August 2006. G. Malandain . Digital Topology. http://www-sop.inria.fr/epidaure/personnel/malandain/topology/. August 2000. IMAGES: Open Problems by R.Klette .  http://www.citr.auckland.ac.nz/dgt/Problem_Files/._Dagstuhl_2004.pdf Tesseract . http://en.wikipedia.org/wiki/Tesseract,  http://en.wikipedia.org/wiki/Image:Changingcube.gif Penteract .  http://mathworld.wolfram.com/images/eps-gif/HypercubeGraphs_850.gif
THANK YOU for listening! Any questions?? ,[object Object]

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Admissible Labelings

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  • 2.
  • 3. Z 2 and Z 3
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  • 5. 2d-Connected M ⊆ Z d 2d-connected if  x,y ∈ M, ∃ 2d-path from x to y in M * analogous for (3 d -1)-connected
  • 6. Digital Topologies on Z d D1 D2 D3 Topologies on Z d satisfying
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  • 10. Stable Sets of Z 2 d M ⊆ Z 2 d where none of its elements are 2d-neighbors Also called stable subsets of the hypercube Q d (vertices are pairwise nonadjacent under Q d ) S := collection of stable sets of Z 2 d
  • 11.
  • 12. Admissible Labelings of Z 2 d (3) means no two points in dom  are 2d-neighbors (3) and (4) imply that  (x) ≤ d-2  x ∈ dom 
  • 13. Admissible Labelings of Z 2 d  1 ,  2 isomorphic if ∃ isometry Φ of Z 2 d onto Z 2 d s.t. dom  1 = Φ( dom  2 ) and  1 =  2 ( Φ ( dom  1 ) )   (0,1)-admissible labeling if  (x) ∈ { 0,1 } otherwise, non-(0,1)-admissible labeling
  • 14. Correspondence isomorphism classes of admissible labelings of Z 2 d homeomorphism classes of digital topologies on Z d (Kong: using normalized admissible functions and digital strict partial orders) 1-1 correspondence
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  • 17. Results for Z 5 Lower bound for number of digital topologies for d=5
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  • 19. References U. Eckhart and L. Latecki . Topologies for the Digital Spaces Z2 and Z3. Computer Vision and Image Understanding 90 (2003) 295-312. R. Klette and A. Rosenfeld . Digital Geometry. Morgan Kauffman Publishers. San Francisco, CA. 2004. T.Y. Kong . The Khalimsky Topologies are precisely those simply connected topologies on Zn whose connected sets include all 2n-connected sets but no (3n-1)-disconnected sets. Theoretical Computer Science 305 (2003) 221-235. T.Y. Kong . Topological adjacency relations on Zn. Theoretical Computer Science 283 (2002) 3-28. T.Y. Kong, R. Kopperman and P.R. Meyer . A Topological Approach to Digital Topology. Amer. Math. Monthly 98 (1991), 901-917. R. Kopperman . The Khalimsky Line as a Foundation for Digital Topology. Shape in Picture (Proceedings of the NATO Advanced Research Workshop held at Driebergen, Spet. 1992). Springer. 1994 V. Kovalevsk y. Axiomatische Digitaltopologie. http://www.bv.inf.tu-dresden.de/AKTUELL/S_KOLL_0406/VORTRAEGE/V1_Kovalevski.ppt. August 2006. G. Malandain . Digital Topology. http://www-sop.inria.fr/epidaure/personnel/malandain/topology/. August 2000. IMAGES: Open Problems by R.Klette . http://www.citr.auckland.ac.nz/dgt/Problem_Files/._Dagstuhl_2004.pdf Tesseract . http://en.wikipedia.org/wiki/Tesseract, http://en.wikipedia.org/wiki/Image:Changingcube.gif Penteract . http://mathworld.wolfram.com/images/eps-gif/HypercubeGraphs_850.gif
  • 20.