• Honam Mathematical Journal
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• v.31 no.3
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• pp.267-278
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• 2009

As a survey-type article, the paper reviews the recent results on a (generalized) universal covering space in digital covering theory. The recent paper [19] established the generalized universal (2, k)-covering property which improves the universal (2, k)-covering property of [3]. In algebraic topology it is well-known that a simply connected and locally path connected covering space is a universal covering space. Unlike this property, in digital covering theory we can propose that a generalized universal covering space has its intrinsic feature. This property can be useful in classifying digital covering spaces and in studying a shortest k-path problem in data structure.

• Honam Mathematical Journal
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• v.32 no.3
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• pp.375-387
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• 2010

The paper studies an existence problem of a (generalized) universal covering space over a digital wedge with a compatible adjacency. In algebraic topology it is well-known that a connected, locally path connected, semilocally simply connected space has a universal covering space. Unlike this property, in digital covering theory we need to investigate its digital version which remains open.

• Honam Mathematical Journal
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• v.31 no.3
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• pp.279-292
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• 2009

As a survey-type article, the paper reviews some results on a regular covering space in digital covering theory. The recent paper [10](see also [12]) established the notion of regular covering space in digital covering theory and studied its various properties. Besides, the papers [14, 16] developed a discrete Deck's transformation group of a digital covering. In this paper we study further their properties. By using these properties, we can classify digital covering spaces. Finally, the paper proposes an open problem.

• Honam Mathematical Journal
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• v.30 no.4
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• pp.589-602
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• 2008

As a survey-type article, the paper reviews various digital topological utilities from digital covering theory. Digital covering theory has strongly contributed to the calculation of the digital k-fundamental group of both a digital space(a set with k-adjacency or digital k-graph) and a digital product. Furthermore, it has been used in classifying digital spaces, establishing almost Van Kampen theory which is the digital version of van Kampen theorem in algebrate topology, developing the generalized universal covering property, and so forth. Finally, we remark on the digital k-surface structure of a Cartesian product of two simple closed $k_i$-curves in ${\mathbf{Z}}^n$, $i{\in}{1,2}$.

• Honam Mathematical Journal
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• v.26 no.1
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• pp.107-117
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• 2004

The aim of this paper is to introduce a digital $({\kappa}_0,\;{\kappa}_1)$-covering map and a local $({\kappa}_0,\;{\kappa}_1)$-homeomorphism. And further, we show that a digital $({\kappa}_0,\;{\kappa}_1)$-covering map is a local $({\kappa}_0,\;{\kappa}_1)$-homeomorphism and the converse does not hold. Finally, some property of a digital covering map is investigated with relation to some restriction map. Furthermore, we raise an open problem with relation to the product covering map.

• Honam Mathematical Journal
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• v.36 no.3
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• pp.695-706
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• 2014

Various properties of digital covering spaces have been substantially used in studying digital homotopic properties of digital images. In particular, these are so related to the study of a digital fundamental group, a classification of digital images, an automorphism group of a digital covering space and so forth. The goal of the present paper, as a survey article, to speak out utility of digital covering theory. Besides, the present paper recalls that the papers [1, 4, 30] took their own approaches into the study of a digital fundamental group. For instance, they consider the digital fundamental group of the special digital image (X, 4), where X := $SC^{2,8}_4$ which is a simple closed 4-curve with eight elements in $Z^2$, as a group which is isomorphic to an infinite cyclic group such as (Z, +). In spite of this approach, they could not propose any digital topological tools to get the result. Namely, the papers [4, 30] consider a simple closed 4 or 8-curve to be a kind of simple closed curve from the viewpoint of a Hausdorff topological structure, i.e. a continuous analogue induced by an algebraic topological approach. However, in digital topology we need to develop a digital topological tool to calculate a digital fundamental group of a given digital space. Finally, the paper [9] firstly developed the notion of a digital covering space and further, the advanced and simplified version was proposed in [21]. Thus the present paper refers the history and the process of calculating a digital fundamental group by using various tools and some utilities of digital covering spaces. Furthermore, we deal with some parts of the preprint [11] which were not published in a journal (see Theorems 4.3 and 4.4). Finally, the paper suggests an efficient process of the calculation of digital fundamental groups of digital images.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.537-545
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• 2005

The aim of this paper is to solve the open problem on product properties of digital covering maps raised from [5]. Namely, let us consider the digital images $X_1 {\subset}Z^{n_{0}}$ with $k_0-adjacency$, $Y_1{\subset}Z^{n_{1}}$ with $k_3-adjacency$, $X_2{\subset}Z^{n_{2}}$ with $k_2-adjacency$ and $Y_2{\subset}Z^{n_{3}}$ with $k_3-adjacency$. Then the reasonable $k_4-adjacency$ of the product image $X_1{\times}X_2$ is determined by the $k_0-$ and $k_2-adjacency$ and the suitable k_5-adjacency$ is assumed on $Y_1{\times}Y_2$ via the $k_1-$ and $k_3-adjacency$ [3] such that each of the projection maps is a digitally continuous map, e.g., $p_1\;:\;X_1{\times}X_2{\rightarrow}X_1$ is a digitally ($k_4,\;k_1$)-continuous map and so on. Let us assume $h_1\;:\;X_1{\rightarrow}Y_1$ to be a digital $(k_0, k_1)$-covering map and $h_2\;:\;X_2{\rightarrow}Y_2$ to be a digital $(k_2,\;k_3)$-covering map. Then we show that the product map $h_1{\times}h_2\;:\;X_1{\times}X_2{\rightarrow}Y_1{\times}Y_2$ need not be a digital $(k_4,k_5)$-covering map.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.207-217
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• 2008

Digital geometry has strongly contributed to the study of a discrete topological space $X{\subset}{\mathbf{Z}}^n$ with k-adjacency of ${\mathbf{Z}}^n$. As a survey-type article, we review various utilities of digital geometry.

• Honam Mathematical Journal
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• v.37 no.1
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• pp.135-147
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• 2015

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.

• Journal of the Korean Mathematical Society
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• v.44 no.6
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• pp.1479-1503
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• 2007

In this paper, we study a strong k-deformation retract derived from a relative k-homotopy and investigate its properties in relation to both a k-homotopic thinning and the k-fundamental group. Moreover, we show that the k-fundamental group of a wedge product of closed k-curves not k-contractible is a free group by the use of some properties of both a strong k-deformation retract and a digital covering. Finally, we write an algorithm for calculating the k-fundamental group of a dosed k-curve by the use of a k-homotopic thinning.