Certain topological methods for computing digital topological complexity
Main Article Content
Abstract
In this paper, we examine the relations of two closely related concepts, the digital Lusternik-Schnirelmann category and the digital higher topological complexity, with each other in digital images. For some certain digital images, we introduce $\kappa-$topological groups in the digital topological manner for having stronger ideas about the digital higher topological complexity. Our aim is to improve the understanding of the digital higher topological complexity. We present examples and counterexamples for $\kappa-$topological groups.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.
References
[1] I. Basabe, J. Gonzalez, Y. Rudyak, and D. Tamaki, Higher topological complexity and its symmetrization, Algebraic Geom. Topol. 14 (2014), 2103–2124. Google Scholar
[2] C. Berge, Graphs and Hypergraphs, 2nd ed. North-Holland, Amsterdam (1976). Google Scholar
[3] A. Borat, and T. Vergili, Digital Lusternik-Schnirelmann category, Turk. J. Math. 42 (2018), 1845–1852. Google Scholar
[4] L. Boxer, Digitally continuous functions, Pattern Recogn. Lett. 15 (1994), 833–839. Google Scholar
[5] L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vis. 10 (1999),51–62. Google Scholar
[6] L. Boxer, Properties of digital homotopy, J. Math. Imaging Vis. 22 (2005), 19–26. Google Scholar
[7] L. Boxer, Homotopy properties of sphere-like digital images, J. Math. Imaging Vis. 24 (2006), 167–175. Google Scholar
[8] L. Boxer, Digital products, wedges, and covering spaces, J. Math. Imaging Vis. 25 (2006), 169–171. Google Scholar
[9] L. Boxer, Continuous maps on digital simple closed curves, Appl. Math. 1 05 (2010), 377–386. Google Scholar
[10] L. Boxer, and I. Karaca, Fundamental groups for digital products, Adv. Appl. Math. Sci. 11(4) (2012), 161–180. Google Scholar
[11] L. Boxer, and P.C. Staecker, Fundamental groups and Euler characteristics of sphere-like digital images, Appl. Gen. Topol. 17(2) (2016), 139–158. Google Scholar
[12] L. Boxer, Alternate product adjacencies in digital topology, Appl. Gen. Topol. 19(1) (2018), 21–53. Google Scholar
[13] O. Ege, and I. Karaca, Digital fibrations, Proc. Nat. Academy Sci. India Sec. A 87 (2017), 109–114. Google Scholar
[14] M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), 211–221. Google Scholar
[15] M. Farber, Invitation to topological robotics, EMS, Zurich (2008). Google Scholar
[16] S.E. Han, Non-product property of the digital fundamental group, Inf. Sci. 171(1-3) (2005), 73–91. Google Scholar
[17] G.T. Herman, Oriented surfaces in digital spaces, CVGIP: Graph. Models Im. Proc. 55 (1993), 381–396. Google Scholar
[18] M. Is, and I. Karaca, The higher topological complexity in digital images, Appl. Gen. Topol. 21 (2020), 305–325. Google Scholar
[19] I. Karaca, and M. Is, Digital topological complexity numbers, Turk. J. Math. 42(6) (2018), 3173–3181. Google Scholar
[20] E. Khalimsky, Motion, deformation, and homotopy in infinite spaces, In: Proceedings IEEE International Conferences on Systems, Man, and Cybernetics. (1987), 227–234. Google Scholar
[21] T.Y. Kong, A digital fundamental group, Comp. Graph. 13 (1989), 159–166. Google Scholar
[22] D.W. Lee, Digital singular homology groups of digital images, Far East J. Math. 88 (2014), 39–63. Google Scholar
[23] D.W. Lee, Digital H-spaces and actions in the pointed digital homotopy category, Appl. Algebra Eng. Comm. Comput. 31 (2020), 149–169. Google Scholar
[24] G. Lupton, J. Oprea, and N. Scoville, Homotopy theory on digital topology, Discrete Comput. Geom. 67 (2021), 112–165. Google Scholar
[25] J.F. Peters, Topology of Digital Images, Visual pattern discovery in proximity spaces (Intelligent Systems Reference Library, 63), Springer (2014). Google Scholar
[26] A. Rosenfeld, Digital topology, Am. Math. Mon. 86 (1979), 76–87. Google Scholar
[27] A. Rosenfeld, ’Continuous’ functions on digital pictures, Pattern Recogn. Lett. 4 (1986), 177–184. Google Scholar
[28] Y. Rudyak, On higher analogs of topological complexity, Topol. Appl. 157(5) (2010), 916–920. Google Scholar
[29] T. Vergili, and A. Borat, Digital Lusternik-Schnirelmann category of digital functions, Hacettepe. J. Math. Stat. 49(4) (2020), 1414–1422. Google Scholar