# Digital Topological Properties of an Alignment of Fixed Point Sets

## Abstract

**:**

## 1. Introduction

- (Q1)
- Given a digital image $(X,k)$, does $F\left(X\right)$ always have ${X}^{\u266f}-1$?Indeed, this issue was partially studied in [4].
- (Q2)
- Given a digital image $(X,k)$, under what condition can we have $F\left(X\right)$ that is perfect?
- (Q3)
- Given a digital image $(X,k)$, how can we explore perfectness of $F\left(X\right)$?
- (Q4)
- For a digital wedge ${C}_{k}^{n,{l}_{1}}\vee {C}_{k}^{n,{l}_{2}}$, assume ${l}_{1}\ge {l}_{2}$. If ${l}_{2}=4$, then is $F({C}_{k}^{n,{l}_{1}}\vee {C}_{k}^{n,4})$ 2-connected?
- (Q5)
- For a digital wedge ${C}_{k}^{n,{l}_{1}}\vee {C}_{k}^{n,{l}_{2}}$, assume ${l}_{1}\ge {l}_{2}$. If ${l}_{2}=6$, then how can we characterize $F({C}_{k}^{n,l}\vee {C}_{k}^{n,6})$?
- (Q6)
- Is $F({C}_{k}^{n,l}\vee {C}_{k}^{n,6})$ perfect?
- (Q7)
- Under what condition is $F({C}_{k}^{n,l}\vee {C}_{k}^{n,6}\vee {C}_{k}^{n,4})$ perfect?

## 2. Preliminaries

- A digital image $(X,k)$ is said to be k-disconnected [15] if there are non-empty sets ${X}_{1},{X}_{2}\subset X$ such that $X={X}_{1}\cup {X}_{2}$, ${X}_{1}\cap {X}_{2}=\varnothing $ and further, there are no points ${x}_{1}\in {X}_{1}$ and ${x}_{2}\in {X}_{2}$ which are k-adjacent. Using this approach, a digital image $(X,k)$ is said to be k-connected (or k-path connected) if it is not k-disconnected.
- A k-connected digital image $(X,k)$ in ${\mathbb{Z}}^{n}$ whose cardinality is greater than 1, the so-called k-path with $l+1$ elements in X is assumed to be the finite sequence ${\left({x}_{i}\right)}_{i\in {[0,l]}_{\mathbb{Z}}}\subset X$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=1$[14]. Then we call the number l the lenth of this k-path.
- We say that a simple k-path is the finite set ${\left({x}_{i}\right)}_{i\in {[0,m]}_{\mathbb{Z}}}\subset {\mathbb{Z}}^{n}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if and only if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=1$[6]. In case ${x}_{0}=x$ and ${x}_{m}=y$, we denote the length of the simple k-path with ${l}_{k}(x,y):=m$.
- A simple closed k-curve (or simple k-cycle) with l elements in ${\mathbb{Z}}^{n}$, denoted by $S{C}_{k}^{n,l}$[11,14], $l\ge 4,l\in {\mathbb{N}}_{0}\setminus \left\{2\right\}$, means the finite set ${\left({x}_{i}\right)}_{i\in {[0,l-1]}_{\mathbb{Z}}}\subset {\mathbb{Z}}^{n}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if and only if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=\pm 1\left(mod\phantom{\rule{0.166667em}{0ex}}l\right)$. Owing to this notion, it is obvious that the number of l should be even. In particular, in the present paper we will use the notation ${C}_{k}^{n,l}$ to abbreviate $S{C}_{k}^{n,l}$. At the moment, we need to remind that in some papers ${C}_{k}^{n,l}$ is used for a notation of a closed k-curve with l elements in ${\mathbb{Z}}^{n}$. However, in this paper since we will not deal with such curves, we may use the notation ${C}_{k}^{n,l}$ for only a simple closed k-curve with l elements in ${\mathbb{Z}}^{n}$.

- The notion of digital wedge (or one point union of two digital images) was initially proposed in [11,16]. To be precise, two given digital images $(X,k)$ and $(Y,k)$, a digital wedge [11,16], denoted by $(X\vee Y,k)$, is defined as the union of the digital images $({X}^{\prime},k)$ and $({Y}^{\prime},k)$, where
- (1)
- ${X}^{\prime}\cap {Y}^{\prime}$ is a singleton, say $\left\{p\right\}$.
- (2)
- ${X}^{\prime}\setminus \left\{p\right\}$ and ${Y}^{\prime}\setminus \left\{p\right\}$ are not k-adjacent, where the two subsets A and B of $(X,k)$ are said to be k-adjacent [14] if $A\cap B=\varnothing $ and there are at least two points $a\in A$ and $b\in B$ such that a is k-adjacent to b.
- (3)
- $({X}^{\prime},k)$ is k-isomorphic to $(X,k)$ and $({Y}^{\prime},k)$ is k-isomorphic to $(Y,k)$ (see Definition 1).

**Proposition 1.**

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

**Definition 1.**

**Definition 2.**

- $(\u20221)$
- for all $x\in X,H(x,0)=f\left(x\right)$ and $H(x,m)=g\left(x\right)$;
- $(\u20222)$
- for all $t\in {[0,m]}_{\mathbb{Z}}$, the induced function ${H}_{t}:X\to Y$ given by ${H}_{t}\left(x\right)=H(x,t)$ for all $x\in X$ is $({k}_{0},{k}_{1})$-continuous.
- $(\u20223)$
- for all $x\in X$, the induced function ${H}_{x}:{[0,m]}_{\mathbb{Z}}\to Y$ given by${H}_{x}\left(t\right)=H(x,t)$ for all $t\in {[0,m]}_{\mathbb{Z}}$ is $(2,{k}_{1})$-continuous;Then we say that H is a $({k}_{0},{k}_{1})$-homotopy between f and g [30].
- $(\u20224)$
- Furthermore, for all $t\in {[0,m]}_{\mathbb{Z}}$, assume that the induced map ${H}_{t}$ on A is a constant which follows the prescribed function from A to Y [11] (see also [33]). To be precise, ${H}_{t}\left(x\right)=f\left(x\right)=g\left(x\right)$ for all $x\in A$ and for all $t\in {[0,m]}_{\mathbb{Z}}$.

**Definition**

**3.**

**Theorem**

**1.**

## 3. Digital Topological Properties of $F({C}_{k}^{n,l}\vee {C}_{k}^{n,4})$ Up to 2-Connectedness and Perfectness of $F({C}_{k}^{n,l}\vee {C}_{k}^{n,4})$

**Definition**

**4.**

**Definition**

**5.**

**Proposition**

**2.**

**Lemma 1.**

**Proof.**

- (1)
- In the case that ${C}_{k}^{n,l}$ is k-contractible, we obviously obtain $l=4$ (see Theorem 1(1)). Hence $F\left({C}_{k}^{n,4}\right)={[0,4]}_{\mathbb{Z}}$. Conversely, since $F\left({C}_{k}^{n,l}\right)={[0,\frac{l}{2}+1]}_{\mathbb{Z}}\cup \left\{l\right\}$[4], by the hypothesis, we obtain that $l=4$, which implies the k-contractibility of ${C}_{k}^{n,l}$.
- (2)
- In the case $l\ge 6$, ${C}_{k}^{n,l}$ is not k-contractible (see Theorem 1). Since $F\left({C}_{k}^{n,l}\right)={[0,\frac{l}{2}+1]}_{\mathbb{Z}}\cup \left\{l\right\}$, we clearly obtain $l-1\notin F\left({C}_{k}^{n,l}\right)$. By using a method similar to the proof of (1), the converse is proved.

**Theorem**

**2.**

**Proof.**

- (a)
- ${f|}_{{C}_{k}^{n,4}}\left(x\right)=x$; or
- (b)
- ${f|}_{{C}_{k}^{n,l}}\left(x\right)=x$; or
- (c)
- $f\left({C}_{k}^{n,l}\right)\u228a{C}_{k}^{n,l}$ and $f\left({C}_{k}^{n,4}\right)\u228a{C}_{k}^{n,4}$ (see Figure 1(1–4)); or
- (d)
- f does not support any fixed point of it, i.e., there is no point $x\in {C}_{k}^{n,l}\vee {C}_{k}^{n,4}$ such that $f\left(x\right)=x$,

- (Case 1)
- If $l=12$, then we see that the number 11 does not belong to $F({C}_{k}^{n,12}\vee {C}_{k}^{n,4})$. To be specific, we obtain$$F({C}_{k}^{n,12}\vee {C}_{k}^{n,4})={[0,10]}_{\mathbb{Z}}\cup {[12,15]}_{\mathbb{Z}}.$$
- (Case 2)
- If $l=14$, then owing to (8), it appears that the elements $12,13$ do not belong to $F({C}_{k}^{n,14}\vee {C}_{k}^{n,4})$. Namely,$$F({C}_{k}^{n,14}\vee {C}_{k}^{n,4})={[0,11]}_{\mathbb{Z}}\cup {[14,17]}_{\mathbb{Z}}.$$
- (Case 3)
- If $l=16$, then we observe that the numbers 13, 14, and 15 do not belong to $F({C}_{k}^{n,16}\vee {C}_{k}^{n,4})$, i.e.,$$F({C}_{k}^{n,16}\vee {C}_{k}^{n,4})={[0,12]}_{\mathbb{Z}}\cup {[16,19]}_{\mathbb{Z}}.$$
- (Case 4)
- If $l=18$, then owing to (8), we obtain that the elements 14, 15, 16, and 17 do not belong to $F({C}_{k}^{n,18}\vee {C}_{k}^{n,4})$, i.e.,$$F({C}_{k}^{n,18}\vee {C}_{k}^{n,4})={[0,13]}_{\mathbb{Z}}\cup {[18,21]}_{\mathbb{Z}}.$$
- (Case 5)
- If $l=20$, then we see that the numbers 15, 16, 17, 18, and 19 do not belong to $F({C}_{k}^{n,20}\vee {C}_{k}^{n,4})$, i.e.,$$F({C}_{k}^{n,l}\vee {C}_{k}^{n,4})={[0,14]}_{\mathbb{Z}}\cup {[20,23]}_{\mathbb{Z}}.$$
- (Case 6)
- If $l=22$, then it appears that the elements $16,17,18,19,20$, and 21 do not belong to $F({C}_{k}^{n,22}\vee {C}_{k}^{n,4})$, i.e.,$$F({C}_{k}^{n,l}\vee {C}_{k}^{n,4})={[0,15]}_{\mathbb{Z}}\cup {[22,25]}_{\mathbb{Z}}.$$

**Remark**

**2.**

**Example 1.**

**Remark**

**3.**

**Theorem**

**3.**

**Proof.**

- (a)
- ${f|}_{{C}_{k}^{n,4}}\left(x\right)=x$; or
- (b)
- ${f|}_{{C}_{k}^{n,l}}\left(x\right)=x$; or
- (c)
- ${f|}_{{C}_{k}^{n,l}\vee {C}_{k}^{n,4}}\left(x\right)=x$; or
- (d)
- ${f|}_{{C}_{k}^{n,4}\vee {C}_{k}^{n,4}}\left(x\right)=x$; or
- (e)
- $f\left({C}_{k}^{n,l}\right)\u228a{C}_{k}^{n,l}$ and $f\left({C}_{k}^{n,4}\right)\u228a{C}_{k}^{n,4}$; or
- (f)
- f does not support any fixed point of it, i.e., there is no $x\in {C}_{k}^{n,l}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4}$ such that $f\left(x\right)=x$.

- (Case 1)
- If $l=12$, then we see that $F({C}_{k}^{n,12}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})={[0,18]}_{\mathbb{Z}}$.
- (Case 2)
- If $l=14$, then we obtain that $F({C}_{k}^{n,14}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})={[0,20]}_{\mathbb{Z}}$ (see Figure 1(5)).
- (Case 3)
- If $l=16$, then we observe that $F({C}_{k}^{n,16}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})={[0,22]}_{\mathbb{Z}}$ (see Figure 1(6)).

**Theorem**

**4.**

- (1)
- $F({C}_{k}^{n,l}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})$ is not 2-connected.
- (2)
- $F({C}_{k}^{n,l}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})$ has two components up to 2-connectedness.

**Proof.**

- (Case 1)
- If $l=18$, then we see that$$F({C}_{k}^{n,18}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})={[0,24]}_{\mathbb{Z}}\setminus \left\{17\right\}.$$
- (Case 2)
- If $l=20$, then we see that$$F({C}_{k}^{n,20}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})={[0,26]}_{\mathbb{Z}}\setminus \{18,19\}.$$
- (Case 3)
- If $l=22$, then we observe that$$F({C}_{k}^{n,22}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})={[0,28]}_{\mathbb{Z}}\setminus \{19,20,21\}\left(see\phantom{\rule{3.33333pt}{0ex}}Figure\phantom{\rule{3.33333pt}{0ex}}1\left(7\right)\right).$$
- (Case 4)
- If $l\ge 24$, then it is obvious that $F({C}_{k}^{n,l}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})$ is not 2-connected.In view of this calculation, if $l\ge 18,l\in {\mathbb{N}}_{0}$, then
- (1)
- $F({C}_{k}^{n,l}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})$ is not perfect, and the proof is completed.
- (2)
- Since the set ${[0,l+6]}_{\mathbb{Z}}\setminus F({C}_{k}^{n,l}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})$ is 2-connected, it appears that $F({C}_{k}^{n,l}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4})$ has only two components.

**Corollary**

**1.**

**Proof.**

- (a)
- ${f|}_{{C}_{k}^{n,4}}\left(x\right)=x$; or
- (b)
- ${f|}_{{C}_{k}^{n,l}}\left(x\right)=x$; or
- (c)
- ${f|}_{{C}_{k}^{n,l}\vee {C}_{k}^{n,4}}\left(x\right)=x$; or
- (d)
- ${f|}_{{C}_{k}^{n,4}\vee {C}_{k}^{n,4}}\left(x\right)=x$; or
- (e)
- ${f|}_{{C}_{k}^{n,l}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4}}\left(x\right)=x$; or
- (f)
- ${f|}_{{C}_{k}^{n,4}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4}}\left(x\right)=x$; or
- (g)
- $f\left({C}_{k}^{n,l}\right)\u228a{C}_{k}^{n,l}$ and $f\left({C}_{k}^{n,4}\right)\u228a{C}_{k}^{n,4}$; or
- (h)
- f does not support any fixed point of it, i.e., there is no $x\in {C}_{k}^{n,l}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4}\vee {C}_{k}^{n,4}$ such that $f\left(x\right)=x$.

**Corollary**

**2.**

**Proof.**

## 4. Non-Perfectness of $F({C}_{k}^{n,l}\vee {C}_{k}^{n,6}),l\ge 6$ and Perfectness of $F({C}_{k}^{n,l}\vee {C}_{k}^{n,6}\vee {C}_{k}^{n,4}),l\in \{4,6,\cdots ,18,20\}$

**Theorem 5.**

**Proof.**

- (a)
- ${f|}_{{C}_{k}^{n,6}}\left(x\right)=x$; or
- (b)
- ${f|}_{{C}_{k}^{n,l}}\left(x\right)=x$; or
- (c)
- $f\left({C}_{k}^{n,l}\right)\u228a{C}_{k}^{n,l}$ and $f\left({C}_{k}^{n,6}\right)\u228a{C}_{k}^{n,6}$; or
- (d)
- f does not support any fixed point of it, i.e., there is no point $x\in {C}_{k}^{n,l}\vee {C}_{k}^{n,6}$ such that $f\left(x\right)=x$,

- (Case 1)
- If $l=6$, then we see that $F({C}_{k}^{n,6}\vee {C}_{k}^{n,6})={[0,9]}_{\mathbb{Z}}\cup \left\{11\right\}$.
- (Case 2)
- If $l=8$, then it appears that $F({C}_{k}^{n,8}\vee {C}_{k}^{n,6})={[0,11]}_{\mathbb{Z}}\cup \left\{13\right\}$.
- (Case 3)
- If $l=10$, then we observe that $F({C}_{k}^{n,10}\vee {C}_{k}^{n,6})={[0,13]}_{\mathbb{Z}}\cup \left\{15\right\}$.
- (Case 4)
- If $l=12$, then we find that $F({C}_{k}^{n,12}\vee {C}_{k}^{n,6})={[0,15]}_{\mathbb{Z}}\cup \left\{17\right\}$.
- (Case 5)
- If $l=14$, then we see that $F({C}_{k}^{n,14}\vee {C}_{k}^{n,6})={[0,17]}_{\mathbb{Z}}\cup \left\{19\right\}$.

**Theorem**

**6.**

**Proof.**

- if $l=16$, then we see that the numbers $15,20$ do not belong to $F({C}_{k}^{n,16}\vee {C}_{k}^{n,6})$.
- If $l=18$, then we see that the numbers $16,17,22$ do not belong to $F({C}_{k}^{n,18}\vee {C}_{k}^{n,6})$.
- If $l=20$, then we see that the numbers $17,18,19,24$ do not belong to $F({C}_{k}^{n,20}\vee {C}_{k}^{n,6})$.

**Theorem**

**7.**

**Proof.**

- (a)
- ${f|}_{{C}_{k}^{n,4}}\left(x\right)=x$;
- (b)
- ${f|}_{{C}_{k}^{n,6}}\left(x\right)=x$;
- (c)
- ${f|}_{{C}_{k}^{n,l}}\left(x\right)=x$;
- (d)
- ${f|}_{{C}_{k}^{n,6}\vee {C}_{k}^{n,4}}\left(x\right)=x$:
- (e)
- ${f|}_{{C}_{k}^{n,l}\vee {C}_{k}^{n,6}}\left(x\right)=x$;
- (f)
- ${f|}_{{C}_{k}^{n,l}\vee {C}_{k}^{n,4}}\left(x\right)=x$;
- (g)
- $f\left({C}_{k}^{n,l}\right)\u228a{C}_{k}^{n,l}$, $f\left({C}_{k}^{n,6}\right)\u228a{C}_{k}^{n,6}$, and $f\left({C}_{k}^{n,4}\right)\u228a{C}_{k}^{n,4}$ (in particular, see Figure 3(5)); and
- (h)
- f does not support any fixed point of it, i.e., there is no point $x\in {C}_{k}^{n,l}\vee {C}_{k}^{n,6}\vee {C}_{k}^{n,4}$ such that $f\left(x\right)=x$,

**Remark**

**4.**

**Example**

**2.**

## 5. Conclusions and Further Work

## Funding

## Conflicts of Interest

## References

- Szymik, M. Homotopies and the universal point property. Order
**2015**, 32, 30–311. [Google Scholar] [CrossRef] [Green Version] - Ege, O.; Karaca, I. Digital homotopy fixed point theory. C. R. Math.
**2015**, 353, 1029–1033. [Google Scholar] [CrossRef] - Han, S.-E. Fixed point property for digital spaces. J. Nonlinear Sci. Appl.
**2017**, 10, 2510–2523. [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. Fixed point set derived form several types of continuities and homotopies in digital topology. Mathematics. submitted.
- Kong, T.Y.; Rosenfeld, A. Digital topology: Introduction and survey. Comput. Vision Graph. Image Process.
**1989**, 48, 357–393. [Google Scholar] [CrossRef] - Kopperman, R. Topological Digital Topology; LNCS: Discrete Geometry for Compurter Imagery; Springer: Berlin, Germany, 2003; Volume 2886, pp. 1–15. [Google Scholar]
- Rosenfeld, A. Digital topology. Am. Math. Mon.
**1979**, 86, 76–87. [Google Scholar] [CrossRef] - Rosenfeld, A. Continuous functions on digital pictures. Pattern Recognit. Lett.
**1986**, 4, 177–184. [Google Scholar] [CrossRef] - Herman, G.T. Oriented surfaces in digital spaces. CVGIP Graph. Model. Image Process.
**1993**, 55, 381–396. [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. The k-homotopic thinning and a torus-like digital image in ℤ
^{n}. J. Math. Imaging Vis.**2008**, 31, 1–16. [Google Scholar] [CrossRef] - Han, S.-E. Estimation of the complexity of a digital image form the viewpoint of fixed point theory. Appl. Math. Comput.
**2019**, 347, 236–248. [Google Scholar] - Kong, T.Y.; Rosenfeld, A. Topological Algorithms for the Digital Image Processing; Elsevier Science: Amsterdam, The Netherlands, 1996. [Google Scholar]
- Han, S.-E. Digital k-Contractibility of an n-times Iterated Connected Sume of Simple Closed k-Surfaces and Almost Fixed Point Property. Mathematics
**2020**, 8, 345. [Google Scholar] [CrossRef] [Green Version] - Han, S.-E. Non-ultra regular digital covering spaces with nontrivial automorphism groups. Filomat
**2013**, 27, 1205–1218. [Google Scholar] [CrossRef] - Berge, C. Graphs and Hypergraphs, 2nd ed.; North-Holland: Amsterdam, The Netherlands, 1976. [Google Scholar]
- Han, S.-E. Strong k-deformation retract and its applications. J. Korean Math. Soc.
**2007**, 44, 1479–1503. [Google Scholar] [CrossRef] [Green Version] - Han, S.-E. Comparison among digital fundamental groups and its applications. Inf. Sci.
**2008**, 178, 2091–2104. [Google Scholar] [CrossRef] - Han, S.-E. Equivalent (k
_{0}, k_{1})-covering and generalized digital lifting. Inf. Sci.**2008**, 178, 550–561. [Google Scholar] [CrossRef] - Han, S.-E. Cartesian product of the universal covering property. Acta Appl. Math.
**2009**, 108, 363–383. [Google Scholar] [CrossRef] - Han, S.-E. Multiplicative property of the digital fundamental group. Acta Appl. Math.
**2010**, 110, 921–944. [Google Scholar] [CrossRef] - Han, S.-E. On the simplicial complex stemmed from a digital graph. Honam Math. J.
**2005**, 27, 115–129. [Google Scholar] - Han, S.-E.; Park, B.G. Digital Graph (k
_{0}, k_{1})-Homotopy Equivalence and Its Applications. 2003. Available online: http://atlas-conferences.com/c/a/k/b/35.htm (accessed on 30 May 2020). - Han, S.-E.; Jafari, S.; Kang, J.M. Topologies on ℤ
^{n}which are not homeomorphic to the n-dimensional Khalimsky topological space. Mathematics**2019**, 7, 1072. [Google Scholar] [CrossRef] [Green Version] - Khalimsky, E. Topological structures in computer sciences. J. Appl. Math. Simul.
**1987**, 1, 25–40. [Google Scholar] [CrossRef] - Wyse, F.; Marcus, D. Solution to problem 5712. Am. Math. Mon.
**1970**, 77, 1119. [Google Scholar] - Alexandorff, P. Diskrete Rume. Mat. Sb.
**1937**, 2, 501–518. [Google Scholar] - Han, S.-E. Digital coverings and their applications. J. Appl. Math. Comput.
**2005**, 18, 487–495. [Google Scholar] - Boxer, L. A classical construction for the digital fundamental group. J. Math. Imaging Vis.
**1999**, 10, 51–62. [Google Scholar] [CrossRef] - Khalimsky, E. Motion, deformation, and homotopy in finite spaces. In Proceedings of the IEEE International Conferences on Systems, Man, and Cybernetics, Boston, MA, USA, 20–23 October 1987; pp. 227–234. [Google Scholar]
- Khalimsky, E. Pattern analysis of n-dimensional digital images. In Proceedings of the IEEE International Conferences on Systems, Man, and Cybernetics, Atlanta, GA, USA, 14–17 October 1986; pp. 1559–1562. [Google Scholar]
- Han, S.-E. Discrete Homotopy of a Closed k-Surface; LNCS 4040; Springer-Verlag: Berlin, Germany, 2006; pp. 214–225. [Google Scholar]
- Han, S.-E. On the classification of the digital images up to a digital homotopy equivalence. J. Comput. Commun. Res.
**2000**, 10, 194–207. [Google Scholar] - Han, S.-E. Fixed point theorems for digital images. Honam Math. J.
**2015**, 37, 595–608. [Google Scholar] [CrossRef] [Green Version] - Han, S.-E. Homotopy equivalence which is suitable for studying Khalimsky nD-spaces. Topol. Appl.
**2012**, 159, 1705–1714. [Google Scholar] [CrossRef] [Green Version] - Han, S.-E. Almost fixed point property for digital spaces associated with Marcus-Wyse topological spaces. J. Nonlinear Sci. Appl.
**2017**, 10, 34–47. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(1) $F({C}_{8}^{2,6}\vee {C}_{8}^{2,4})={[0,9]}_{\mathbb{Z}}$; (2) $F({C}_{8}^{2,8}\vee {C}_{8}^{2,4})={[0,11]}_{\mathbb{Z}}$; (3) $F({C}_{8}^{2,10}\vee {C}_{8}^{2,4})={[0,13]}_{\mathbb{Z}}$; (4) $F({C}_{8}^{2,12}\vee {C}_{8}^{2,4})={[0,15]}_{\mathbb{Z}}\setminus \left\{11\right\}$; (5) $F({C}_{8}^{2,14}\vee {C}_{8}^{2,4}\vee {C}_{8}^{2,4})={[0,20]}_{\mathbb{Z}}$; (6) $F({C}_{8}^{2,16}\vee {C}_{8}^{2,4}\vee {C}_{8}^{2,4})={[0,22]}_{\mathbb{Z}}$; (7) $F({C}_{8}^{2,22}\vee {C}_{8}^{2,4}\vee {C}_{8}^{2,4})={[0,28]}_{\mathbb{Z}}\setminus \{19,20,21\}$; and (8) $F({C}_{8}^{2,18}\vee {C}_{8}^{2,4}\vee {C}_{8}^{2,4}\vee {C}_{8}^{2,4})={[0,27]}_{\mathbb{Z}}$.

**Figure 2.**In these figures, the dotted arrows indicate 8-continuous mappings in the $DTC\left(8\right)$-setting. (1) Perfectness of $F(X,8)$; (2) Perfectness of $F({C}_{8}^{2,4}\vee {C}_{8}^{2,4})$.

**Figure 3.**In these figures, the dotted arrows indicate 8-continuous self-mappings. (1)–(4) Explanation of being two components of $F({C}_{k}^{n,l}\vee {C}_{k}^{n,6})$, $l\in \{8,10,12,14\}$; (5) Explanation of the perfectness of $F({C}_{k}^{n,l}\vee {C}_{k}^{n,6}\vee {C}_{k}^{n,4})={[0,l+8]}_{\mathbb{Z}}$, $l\in \{4,6,8,10,12,14,16,18,20\}$ (see Theorem 7). In particular, $l=14$.

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Han, S.-E.
Digital Topological Properties of an Alignment of Fixed Point Sets. *Mathematics* **2020**, *8*, 921.
https://doi.org/10.3390/math8060921

**AMA Style**

Han S-E.
Digital Topological Properties of an Alignment of Fixed Point Sets. *Mathematics*. 2020; 8(6):921.
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**Chicago/Turabian Style**

Han, Sang-Eon.
2020. "Digital Topological Properties of an Alignment of Fixed Point Sets" *Mathematics* 8, no. 6: 921.
https://doi.org/10.3390/math8060921