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2021 | OriginalPaper | Chapter

A Convenient Graph Connectedness for Digital Imagery

Author : Josef Šlapal

Published in: High Performance Computing in Science and Engineering

Publisher: Springer International Publishing

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Abstract

In a simple undirected graph, we introduce a special connectedness induced by a set of paths of length 2. We focus on the 8-adjacency graph (with the vertex set \(\mathbb {Z}^2\)) and study the connectedness induced by a certain set of paths of length 2 in the graph. For this connectedness, we prove a digital Jordan curve theorem by determining the Jordan curves, i.e., the circles in the graph that separate \(\mathbb {Z}^2\) into exactly two connected components. These Jordan curves are shown to have an advantage over those given by the Khalimsky topology on \(\mathbb {Z}^2\).

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previous chapter Analysis and Visualization of the Dynamic Behavior of HPC Applications

Chvátal, V.: Correction to: a de Bruijn-Erdős theorem in graphs? In: Gera, R., Haynes, T.W., Hedetniemi, S.T. (eds.) Graph Theory. PBM, pp. C1–C2. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-97686-0_15CrossRef

Engelking, R.: General Topology. Państwowe Wydawnictwo Naukowe, Warszawa (1977)MATH

Khalimsky, E.D., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topology Appl. 36, 1–17 (1990)MathSciNetCrossRef

Khalimsky, E.D., Kopperman, R., Meyer, P.R.: Boundaries in digital plane. J. Appl. Math. Stochast. Anal. 3, 27–55 (1990) MathSciNetCrossRef

Kiselman, C.O.: Digital jordan curve theorems. In: Borgefors, G., Nyström, I., di Baja, G.S. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 46–56. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44438-6_5CrossRef

Kong, T.Y., Kopperman, R., Meyer, P.: A topological approach to digital topology. Amer. Math. Monthly 98, 902–917 (1991)MathSciNetCrossRef

Kong, T.Y., Roscoe, W.: A theory of binary digital pictures. Comput. Vision Graphics Image Proc. 32, 221–243 (1985)CrossRef

Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vision Graphics Image Proc. 48, 357–393 (1989)CrossRef

Kopperman, R., Meyer, P.R., Wilson, R.G.: A Jordan surface theorem for three-dimensional digital space. Discrete Comput. Geom. 6, 155–161 (1991)MathSciNetCrossRef

10.

Melin, E.: Digital surfaces and boundaries in Khalimsky spaces. J. Math. Imaging Vision 28, 169–177 (2007)MathSciNetCrossRef

11.

Melin, E.: Continuous digitization in Khalimsky spaces. J. Approx. Theory 150, 96–116 (2008)MathSciNetCrossRef

12.

Rosenfeld, A.: Connectivity in digital pictures. J. Assoc. Comput. Math. 17, 146–160 (1970)MathSciNetCrossRef

13.

Rosenfeld, A.: Digital topology. Amer. Math. Monthly 86, 621–630 (1979)MathSciNetCrossRef

14.

Rosenfeld, A.: Picture Languages. Academic Press, New York (1979)MATH

15.

Šlapal, J.: Jordan curve theorems with respect to certain pretopologies on \(\mathbb{Z}^2\). In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 252–262. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04397-0_22CrossRef

16.

Slapal, J.: A jordan curve theorem in the digital plane. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds.) IWCIA 2011. LNCS, vol. 6636, pp. 120–131. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21073-0_13CrossRef

17.

Šlapal, J.: Graphs with a path partition for structuring digital spaces. Inform. Sci. 233, 305–312 (2013)MathSciNetCrossRef

18.

Šlapal, J.: Convenient adjacencies on \(\mathbb{Z}^2\). Filomat 28, 305–312 (2014)MathSciNetCrossRef

Title: A Convenient Graph Connectedness for Digital Imagery
Author: Josef Šlapal
Publisher: Springer International Publishing
Book: High Performance Computing in Science and Engineering
Print ISBN: 978-3-030-67076-4

Electronic ISBN: 978-3-030-67077-1

Copyright Year: 2021
DOI: https://doi.org/10.1007/978-3-030-67077-1_9

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