# Fixed Point Sets of Digital Curves and Digital Surfaces

## Abstract

**:**

## 1. Introduction

## 2. Digital Wedges and Some Properties of the Digital Continuity

**Proposition**

**1**

**Theorem**

**1.**

**Proof.**

- The set of $(X,k)$, where $X\subset {\mathbb{Z}}^{n}$, as objects of DTC denoted by $Ob(DTC)$;
- For every ordered pair of objects $({X}_{i},{k}_{i}),i\phantom{\rule{3.33333pt}{0ex}}\in \{0,1\}$, the set of all $({k}_{0},{k}_{1})$-continuous maps between them as morphisms of DTC, denoted by $Mor(DTC)$.

**Definition**

**1**

- (1)
- ${A}^{\prime}\cap {B}^{\prime}$ is a singleton, say $\left\{p\right\}$;
- (2)
- ${A}^{\prime}\backslash \left\{p\right\}$ and ${B}^{\prime}\backslash \left\{p\right\}$ are not k-adjacent, where two sets $(C,k)$ and $(D,k)$ are said to be k-adjacent if $C\cap D=\varnothing $ and there are at least two points $a\in C$ and $b\in D$ such that a is k-adjacent to b [20]; and
- (3)
- $({A}^{\prime},k)$ is k-isomorphic to $(A,k)$ and $({B}^{\prime},k)$ is k-isomorphic to $(B,k)$ (see Definition 1).

## 3. Formulation of $\mathit{F}(\mathit{C}\mathit{o}{\mathit{n}}_{\mathit{k}}({\mathit{C}}_{\mathit{k}}^{\mathit{n},\mathit{l}}\vee {\mathit{C}}_{\mathit{k}}^{\mathit{n},4})),\mathit{l}\phantom{\rule{3.33333pt}{0ex}}\in {\mathbb{N}}_{1}\backslash \{1,3,5\},\mathit{k}\phantom{\rule{3.33333pt}{0ex}}\ne 2\mathit{n}$ and Its Digital Topological Properties

**Definition**

**2**

**Definition**

**3**

**Lemma**

**1.**

**Remark**

**1.**

**Theorem**

**3.**

**Proof.**

- (a)
- ${f|}_{{C}_{k}^{n,4}}(x)=x$; or
- (b)
- ${f|}_{{C}_{k}^{n,l}}(x)=x$; or
- (c)
- $f({C}_{k}^{n,l})\u228a{C}_{k}^{n,l}$ and $f({C}_{k}^{n,4})\u228a{C}_{k}^{n,4}$; or
- (d)
- f does not support any fixed point of ${C}_{k}^{n,l}\vee {C}_{k}^{n,4}$.

**Corollary**

**1.**

**Proof.**

**Example**

**1.**

- (a)
- $F(Co{n}_{8}({C}_{8}^{2,7}\vee {C}_{8}^{2,4}))={[0,10]}_{\mathbb{Z}}$.Similarly, we obtain the following (see Figure 3a,b).
- (b)
- $F(Co{n}_{8}({C}_{8}^{2,9}\vee {C}_{8}^{2,4}))={[0,12]}_{\mathbb{Z}}$ (see Figure 3a).
- (c)
- $F(Co{n}_{18}({C}_{18}^{3,7}\vee {C}_{18}^{3,4}))={[0,10]}_{\mathbb{Z}}$ (see Figure 3b).

**Remark**

**2.**

**Remark**

**3.**

**Proof.**

**Lemma**

**2**

**Theorem**

**4**

**Theorem**

**5.**

- (1)
- In the case $l\in {\mathbb{N}}_{0},F(Co{n}_{k}({C}_{k}^{n,l}\vee {C}_{k}^{n,4}))$ is perfect if and only if $l\in \{4,6,8,10\}$ [3].
- (2)
- In the case $l\in {\mathbb{N}}_{1},k=8,n=2$, $F(Co{n}_{k}({C}_{k}^{n,l}\vee {C}_{k}^{n,4}))$ is perfect if and only if $l\in \{7,9\}$.
- (3)
- In the case $l\in {\mathbb{N}}_{1},k=18,n=3$, $F(Co{n}_{k}({C}_{k}^{n,l}\vee {C}_{k}^{n,4}))$ is perfect if and only if $l\in \{7,9\}$.
- (4)
- In the case $l\in {\mathbb{N}}_{1},k=26,n=3$, $F(Co{n}_{k}({C}_{k}^{n,l}\vee {C}_{k}^{n,4}))$ is perfect if and only if $l\in \{5,7,9\}$.

**Proof.**

## 4. Alignments of Fixed Point Sets of ${\mathit{C}}_{\mathit{k}}^{\mathit{n},{\mathit{l}}_{1}}\vee {\mathit{C}}_{\mathit{k}}^{\mathit{n},{\mathit{l}}_{2}},{\mathit{l}}_{1}\in {\mathbb{N}}_{1},{\mathit{l}}_{2}\in {\mathbb{N}}_{0},\mathit{k}\phantom{\rule{3.33333pt}{0ex}}\ne 2\mathit{n}$

**Theorem**

**6.**

**Proof.**

**Example 2.**

- (1)
- $F(Co{n}_{k}({C}_{k}^{n,13}\vee {C}_{k}^{n,6}))={[0,16]}_{\mathbb{Z}}\cup \left\{18\right\}$.
- (2)
- $F(Co{n}_{k}({C}_{k}^{n,15}\vee {C}_{k}^{n,6}))={[0,13]}_{\mathbb{Z}}\cup {[15,18]}_{\mathbb{Z}}\cup \left\{20\right\}$.
- (3)
- $F(Co{n}_{k}({C}_{k}^{n,21}\vee {C}_{k}^{n,6}))={[0,16]}_{\mathbb{Z}}\cup {[21,24]}_{\mathbb{Z}}\cup \left\{26\right\}$.

**Remark**

**4.**

**Theorem**

**7.**

- (1)
- $F(Co{n}_{k}({C}_{k}^{n,{l}_{1}}\vee {C}_{k}^{n,{l}_{2}}))$ has three 2-components if and only if ${l}_{1}\ge 2{l}_{2}+3$.
- (2)
- $F(Co{n}_{k}({C}_{k}^{n,{l}_{1}}\vee {C}_{k}^{n,{l}_{2}}))$ has two 2-components if and only if ${l}_{1}\le 2{l}_{2}+1$.

**Proof.**

**Example**

**3.**

**Corollary**

**2.**

**Remark**

**5.**

**Remark**

**6.**

**Theorem**

**8.**

**Proof.**

**Example**

**4.**

- (1)
- $F(Co{n}_{k}({C}_{k}^{n,13}\vee {C}_{k}^{n,6}\vee {P}_{1})={[0,19]}_{\mathbb{Z}}$ which is perfect, where ${P}_{1}$ is a simple k-path with length 1 (one).
- (2)
- $F(Co{n}_{k}({C}_{k}^{n,15}\vee {C}_{k}^{n,6}\vee {P}_{1}))={[0,21]}_{\mathbb{Z}}$ which is perfect, where ${P}_{1}$ is a simple k-path with length 1 (one).

**Theorem**

**9.**

**Proof.**

**Example 5.**

- (1)
- $F(Co{n}_{k}({C}_{k}^{n,13}\vee {C}_{k}^{n,6}\vee {C}_{k}^{n,4}))={[0,21]}_{\mathbb{Z}}$, which is perfect.
- (2)
- $F(Co{n}_{k}({C}_{k}^{n,15}\vee {C}_{k}^{n,6}\vee {C}_{k}^{n,4}))={[0,23]}_{\mathbb{Z}}$.

## 5. Digital Topological Properties of Alignments of Fixed Point Sets of ${\mathit{C}}_{\mathit{k}}^{\mathit{n},{\mathit{l}}_{\mathbf{1}}}\vee {\mathit{C}}_{\mathit{k}}^{\mathit{n},{\mathit{l}}_{\mathbf{2}}},{\mathit{l}}_{\mathbf{1}},{\mathit{l}}_{\mathbf{2}}(\ge \mathbf{7})\in {\mathbb{N}}_{\mathbf{1}},\mathit{k}\phantom{\rule{3.33333pt}{0ex}}\ne \mathbf{2}\mathit{n}$

**Theorem**

**10.**

**Proof.**

**Example 6.**

- (1)
- $F(Co{n}_{k}({C}_{k}^{n,13}\vee {C}_{k}^{n,7}))={[0,16]}_{\mathbb{Z}}\cup \left\{19\right\}$.
- (2)
- $F(Co{n}_{k}({C}_{k}^{n,15}\vee {C}_{k}^{n,7}))={[0,18]}_{\mathbb{Z}}\cup \left\{21\right\}$.
- (3)
- $F(Co{n}_{k}({C}_{k}^{n,21}\vee {C}_{k}^{n,7}))={[0,17]}_{\mathbb{Z}}\cup {[21,24]}_{\mathbb{Z}}\cup \left\{27\right\}$.

**Remark**

**7.**

**Theorem**

**11.**

- (1)
- $F(Co{n}_{k}({C}_{k}^{n,{l}_{1}}\vee {C}_{k}^{n,{l}_{2}}))$ has three 2-components if and only if ${l}_{1}\ge 2{l}_{2}+3$.
- (2)
- $F(Co{n}_{k}({C}_{k}^{n,{l}_{1}}\vee {C}_{k}^{n,{l}_{2}}))$ has two 2-components if and only if ${l}_{1}\le 2{l}_{2}+1$.

**Proof.**

**Example**

**7.**

- (1)
- $F(Co{n}_{k}({C}_{k}^{n,13}\vee {C}_{k}^{n,7})$ has two 2-components.
- (2)
- $F(Co{n}_{k}({C}_{k}^{n,15}\vee {C}_{k}^{n,7})$ has two 2-components.
- (3)
- $F(Co{n}_{k}({C}_{k}^{n,21}\vee {C}_{k}^{n,7}))$ has three 2-components.

**Remark**

**8.**

**Remark 9.**

- (1)
- $F(Co{n}_{k}({C}_{k}^{n,5}\vee {C}_{k}^{n,5}))={[0,7]}_{\mathbb{Z}}\cup \left\{9\right\}$.
- (2)
- Given ${C}_{k}^{n,5}\vee {C}_{k}^{n,l}$, $l(\ge 5)\in {\mathbb{N}}_{1}$, we obtain $F(Co{n}_{k}({C}_{k}^{n,5}\vee {C}_{k}^{n,l}))={[0,\frac{l+9}{2}]}_{\mathbb{Z}}\cup {[l,l+2]}_{\mathbb{Z}}\cup \{l+4\}$, which is not perfect.

**Theorem**

**12.**

**Proof.**

**Example 8.**

- (1)
- $F(Co{n}_{k}({C}_{k}^{n,15}\vee {C}_{k}^{n,7}\vee {P}_{2}))={[0,23]}_{\mathbb{Z}}$, where ${P}_{2}$ is a simple k-path with length 2.
- (2)
- $F(Co{n}_{k}({C}_{k}^{n,21}\vee {C}_{k}^{n,7}\vee {P}_{3}))={[0,30]}_{\mathbb{Z}}$, where ${P}_{3}$ is a simple k-path with length 3.

**Theorem**

**13.**

**Proof.**

**Example**

**9.**

## 6. Digital Topological Properties of Alignments of Digital $\mathit{k}$-Surfaces

**Definition**

**4**

- (1)
- $MS{S}_{6}(\subset {\mathbb{Z}}^{3})$ is 6-isomorphic to $(X,6)$, where $X:={[-1,1]}_{\mathbb{Z}}^{3}\backslash \left\{{0}_{3}\right\}$, i.e., $MS{S}_{6}{\approx}_{6}{[-1,1]}_{\mathbb{Z}}^{3}\backslash \left\{{0}_{3}\right\}$, where ${0}_{3}:=(0,0,0)$ (see Figure 4a).
- (2)
- $MS{S}_{18}^{\prime}{\approx}_{18}(Y,18)$, where $Y:=\{p\in {\mathbb{Z}}^{3}|d(p,{0}_{3})=1\}$ (see Figure 5b(1)), d is the Euclidean distance in ${\mathbb{R}}^{3}$.
- (3)
- (4)
- $MS{S}_{26}^{\prime}:=MS{S}_{18}^{\prime}$.

**Remark**

**10**

- (1)
- $MS{S}_{6}$ is not 6-contractible.
- (2)
- $MS{S}_{18}^{\prime}$ and $MS{S}_{18}$ are considered in the digital pictures $({\mathbb{Z}}^{3},18,6,MS{S}_{18}^{\prime})$ and $({\mathbb{Z}}^{3},18,6,MS{S}_{18})$, respectively. Besides, each of them is 18-contractible.
- (3)

**Theorem**

**14.**

**Proof.**

**Theorem 15.**

- (1)
- $F(Co{n}_{18}(MS{S}_{18}))$ is not perfect, i.e., $F(Co{n}_{18}(MS{S}_{18}))={[0,8]}_{\mathbb{Z}}\cup \left\{10\right\}$, which has two 2-components.
- (2)
- $F(Co{n}_{18}(MS{S}_{18}^{\prime}))$ is perfect.
- (3)
- $F(Co{n}_{26}(MS{S}_{26}^{\prime}))$ is perfect.

**Proof.**

**Remark**

**11.**

**Theorem**

**16.**

**Proof.**

**Theorem**

**17.**

**Proof.**

**Corollary**

**3.**

**Proof.**

## 7. Conclusions

## Funding

## Conflicts of Interest

## References

- Szymik, M. Homotopies and the universal point property. Order
**2015**, 32, 30–311. [Google Scholar] [CrossRef] [Green Version] - Boxer, L.; Staecker, P.C. Fixed point sets in digital topology, 1. Appl. Gen. Topol.
**2020**, 21, 87–110. [Google Scholar] [CrossRef] [Green Version] - Han, S.-E. Digital topological properties of an alignment of fixed point sets. Mathematics
**2020**, 8, 921. [Google Scholar] [CrossRef] - Han, S.-E. Fixed point sets of k-continuous self-maps of m-iterated digital wedges. Mathematics
**2020**, 8, 1617. [Google Scholar] [CrossRef] - El-Sabaa, F.; El-Tarazi, M. The chaotic motion of a rigid body roating about a fixed point. In Predictability, Stability, and in N-Body Dynamical Systems, NATO ASI Series, NSSB; Plenum Press: New York, NY, USA, 1991; Volume 272, pp. 573–581. [Google Scholar]
- Berge, C. Graphs and Hypergraphs, 2nd ed.; North-Holland: Amsterdam, The Netherlands, 1976. [Google Scholar]
- Bertrand, G. Simple points, topological numbers and geodesic neighborhoods in cubic grids. Pattern Recognit. Lett.
**1994**, 15, 1003–1011. [Google Scholar] [CrossRef] - Bertrand, G.; Malgouyres, M. Some topological properties of discrete surfaces. J. Math. Imaging Vis.
**1999**, 20, 207–221. [Google Scholar] [CrossRef] - Chen, L. Digital and Discrete Geometry, Theory and Algorithm; Springer: Berlin/Heidelberg, Germany, 2014; ISBN 978-3-319-12098-0. [Google Scholar]
- Han, S.-E. Minimal simple closed 18-surfaces and a topological preservation of 3D surfaces. Inf. Sci.
**2006**, 176, 120–134. [Google Scholar] [CrossRef] - Han, S.-E. Digital k-Contractibility of an n-times Iterated Connected Sum of Simple Closed k-Surfaces and Almost Fixed Point Property. Mathematics
**2020**, 8, 345. [Google Scholar] [CrossRef] [Green Version] - Malgouyres, R.; Lenoir, A. Topology preservation within digital surfaces. Graph. Model.
**2000**, 62, 71–84. [Google Scholar] [CrossRef] [Green Version] - Kong, T.Y.; Rosenfeld, A. Topological Algorithms for the Digital Image Processing; Elsevier Science: Amsterdam, The Netherlands, 1996. [Google Scholar]
- Rosenfeld, A. Digital topology. Am. Math. Monthly
**1976**, 86, 76–87. [Google Scholar] - Rosenfeld, A. Continuous functions on digital pictures. Pattern Recognit. Lett.
**1986**, 4, 177–184. [Google Scholar] [CrossRef] - Han, S.-E. Non-product property of the digital fundamental group. Inf. Sci.
**2005**, 171, 73–92. [Google Scholar] [CrossRef] - Han, S.-E. Estimation of the complexity of a digital image form the viewpoint of fixed point theory. J. Appl. Math. Comput.
**2019**, 347, 236–248. [Google Scholar] [CrossRef] - Han, S.-E. Non-ultra regular digital covering spaces with nontrivial automorphism groups. Filomat
**2013**, 27, 1205–1218. [Google Scholar] [CrossRef] - Herman, G.T. Oriented surfaces in digital spaces. CVGIP Graph. Model. Image Process.
**1993**, 55, 381–396. [Google Scholar] [CrossRef] - Kong, T.Y.; Rosenfeld, A. Digital topology: Introduction and survey. Comput. Vision Graph. Image Process.
**1989**, 48, 357–393. [Google Scholar] [CrossRef] - Kong, T.Y.; Roscoe, A. Continuous analogs of axiomatized digital surfaces. Comput. Vision Graph. Image Process.
**1985**, 29, 60–85. [Google Scholar] [CrossRef] - Han, S.-E. On the simplicial complex stemmed from a digital graph. Honam Math. J.
**2005**, 27, 115–129. [Google Scholar] - Boxer, L. A classical construction for the digital fundamental group. Math. Imaging Vis.
**1999**, 10, 51–62. [Google Scholar] [CrossRef] - Munkres, J.R. Topology A First Course; Prentice-Hall, Inc.: Upper Saddle River, NJ, USA, 1975. [Google Scholar]
- Shee, S.-C.; Ho, Y.-S. The cordiality of one point union of n-copies of a graph. Discret. Math.
**1993**, 117, 225–243. [Google Scholar] [CrossRef] [Green Version] - Han, S.-E. Fixed point theorems for digital images. Honam Math. J.
**2015**, 37, 595–608. [Google Scholar] [CrossRef] [Green Version] - Morgenthaler, D.G.; Rosenfeld, A. Surfaces in three dimensional digital images. Inf. Control.
**1981**, 51, 227–247. [Google Scholar] [CrossRef] [Green Version] - Peters, J.F. Computational Geometry, Topology and Physics of Digital Images with Applications. Shape Complexes, Optical Vortex Nerves and Proximities; Springer Nature: Cham, Switzerland, 2020; p. xxv+440. [Google Scholar]
- Kiselman, C.O. Digital Geometry and Mathematical Morphology. Lecture Notes. Uppsala University, Department of Mathematics, 2002. Available online: www.math.uu.se/~kiselman (accessed on 30 July 2020).
- Dolhare, U.P.; Nalawade, V.V. Fixed point theorems in digital images and applications to fractal image compression. Asian J. Math. Comput. Res.
**2018**, 25, 18–37. [Google Scholar] - Verkhovod, Y.V.; Gorr, G.V. Precessional-isoconic motion of a rigid body with a fixed point. J. Appl. Math. Mech.
**1993**, 57, 613–622. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) One specific example of a member of the set of closed curves, ${C}_{8}^{2,7}$ [4]. Here, $n=2$, the underlying 2-dimensional lattice is shown as a dashed grid. The closed curve consisting of 7 points appears black dots labeled ${x}_{1}$ through ${x}_{6}$. (

**b**) One specific example of a member of the set of closed curves, ${C}_{18}^{3,9}$. Now $n=3$, so the lattice is 3-dimensional (dashed grid). The closed curve of 9 points runs from 0 through 8.

**Figure 2.**Comparison between the $k({t}_{1},n)$- and $k({t}_{2},n)$-continuities, ${t}_{1}\u2a87{t}_{2}$, which supports the proof of Theorem 1. (

**a**) 26-continuity of ${f}_{1}$ need not imply 18-continuity of it. (

**b**) 6-continuity of ${f}_{2}$ need not imply k-continuity of it, $k\in \{18,26\}$. (

**c**) 18-continuity of ${f}_{3}$ need not imply 26-continuity of it.

**Figure 3.**(

**a**) Configuration of $F(Co{n}_{8}({C}_{8}^{2,9}\vee {C}_{8}^{2,4}))={[0,12]}_{\mathbb{Z}}$. (

**b**) For $F(Co{n}_{18}({C}_{18}^{3,7}\vee {C}_{18}^{3,4}))={[0,10]}_{\mathbb{Z}}$.

**Figure 4.**Configuration of 6-continuous self-maps ${h}_{i}$ of $MS{S}_{6}$ such that $Fix{({h}_{i})}^{\u266f}=i,i\phantom{\rule{3.33333pt}{0ex}}\in {[0,17]}_{\mathbb{Z}}$. (

**a**) $MS{S}_{6}$. (

**b**) The image by the self-map ${h}_{17}$ of $MS{S}_{6}$. (

**c**) The image by the self-map ${h}_{16}$ of $MS{S}_{6}$. (

**d**) The image by the self-map ${h}_{15}$ of $MS{S}_{6}$. (

**e**) The image by the self-map ${h}_{14}$ of $MS{S}_{6}$. This functions support the maps in the proof of Theorem 14.

**Figure 5.**(

**a**) Configuration of 18-continuous self-maps ${f}_{i}$ of $MS{S}_{18}$ such that $Fix{({f}_{i})}^{\u266f}=i,i\phantom{\rule{3.33333pt}{0ex}}\in {[0,8]}_{\mathbb{Z}}$. (

**b**) Description of 18-continuous self-maps ${g}_{i}$ of $MS{S}_{18}^{\prime}$ such that $Fix{({g}_{i})}^{\u266f}=i,i\phantom{\rule{3.33333pt}{0ex}}\in {[0,5]}_{\mathbb{Z}}$.

**Figure 6.**(

**a**,

**b**) Explanation of the process of establishing $MS{S}_{18}\vee MS{S}_{18}^{\prime}$. (

**c**) $MS{S}_{18}\vee MS{S}_{18}\vee MS{S}_{18}^{\prime}$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Han, S.-E.
Fixed Point Sets of Digital Curves and Digital Surfaces. *Mathematics* **2020**, *8*, 1896.
https://doi.org/10.3390/math8111896

**AMA Style**

Han S-E.
Fixed Point Sets of Digital Curves and Digital Surfaces. *Mathematics*. 2020; 8(11):1896.
https://doi.org/10.3390/math8111896

**Chicago/Turabian Style**

Han, Sang-Eon.
2020. "Fixed Point Sets of Digital Curves and Digital Surfaces" *Mathematics* 8, no. 11: 1896.
https://doi.org/10.3390/math8111896