Abstract
We study closure operators on graphs which are induced by path partitions, i.e., certain sets of paths of the same lengths in these graphs. We investigate connectedness with respect to the closure operators studied. In particular, the closure operators are discussed that are induced by path partitions of some natural graphs on the digital spaces \({\mathbb {Z}}^n\), \(n>0\) a natural number. For the case \(n=2\), i.e., for the digital plane \({\mathbb {Z}}^2\), the induced closure operators are shown to satisfy an analogue of the Jordan curve theorem, which allows using them as convenient background structures for studying digital images.
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References
Čech, E.: Topological spaces. In: Topological Papers of Eduard Čech, pp. 436–472. Academia, Prague (1968)
Čech, E.: Topological Spaces. Academia, Prague (1966). (revised by Z. Frolík and M. Katětov)
Engelking, R.: General Topology. Państwowe Wydawnictwo Naukowe, Warszawa (1977)
Harrary, F.: Graph Theory. Addison-Wesley Publ. Comp., Reading (1969)
Khalimsky, E.D., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topol. Appl. 36, 1–17 (1990)
Kong, T.Y., Roscoe, W.: A theory of binary digital pictures. Comput. Vis. Graph. Image Process. 32, 221–243 (1985)
Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48, 357–393 (1989)
Kong, T.Y., Kopperman, R., Meyer, P.R.: A topological approach to digital topology. Am. Math. Mon. 98, 902–917 (1991)
Rosenfeld, A.: Connectivity in digital pictures. J. Assoc. Comput. Mach. 17, 146–160 (1970)
Rosenfeld, A.: Digital topology. Am. Math. Mon. 86, 621–630 (1979)
Šlapal, J.: Direct arithmetics of relational systems. Publ. Math. Debr. 38, 39–48 (1991)
Šlapal, J.: A digital analogue of the Jordan curve theorem. Discret. Appl. Math. 139, 231–251 (2004)
Šlapal, J.: Convenient closure operators on \(\mathbb{Z}^2\). In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 425–436. Springer, Heidelberg (2009). doi:10.1007/978-3-642-10210-3_33
Šlapal, J.: A quotient universal digital topology. Theor. Comput. Sci. 405, 164–175 (2008)
Šlapal, J.: Graphs with a walk partition for structuring digital spaces. Inf. Sci. 233, 305–312 (2013)
Acknowledgement
This work was supported by Ministry of Education, Youth and Sports of the Czech Republic from the National Programme of Sustainability (NPU II) project “IT4Innovations Excellence in Science - LQ1602”.
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Šlapal, J. (2017). Structuring Digital Spaces by Path-Partition Induced Closure Operators on Graphs. In: Barneva, R., Brimkov, V., Tavares, J. (eds) Computational Modeling of Objects Presented in Images. Fundamentals, Methods, and Applications. CompIMAGE 2016. Lecture Notes in Computer Science(), vol 10149. Springer, Cham. https://doi.org/10.1007/978-3-319-54609-4_3
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DOI: https://doi.org/10.1007/978-3-319-54609-4_3
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