# Fixed Point Theory for Digital k-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces

## Abstract

**:**

^{n}− 1)-neighborhood; digital topology

## 1. Introduction

- (Q1)
- How to establish a geometric realization of an ${S}_{k}$?
- (Q2)
- Does the geometric realization transform an ${S}_{k}$ into a certain spherical (or a sphere-like) polyhedron in ${\mathbb{R}}^{3}$?
- (Q3)
- How to define the Euler characteristic of an ${S}_{k}$?
- (Q4)
- Are there certain relationships between the Euler characteristic of an ${S}_{k}$ and that of a geometric realization of an ${S}_{k}$?
- (Q5)
- What about the FPP or the AFPP for an ${S}_{k}$?

## 2. Basic Notions Related to Digital $\mathit{k}$-Surfaces and a Connected Sum for $\mathit{k}$-Surfaces

- We say that two subsets $(A,k)$ and $(B,k)$ of $(X,k)$ are k-adjacent if $A\cap B=\varnothing $ and that there are points $a\in A$ and $b\in B$ such that a and b are k-adjacent [17]. In particular, in case B is a singleton, say $B=\{x\}$, we say that A is k-adjacent to x.
- For a k-adjacency relation of ${\mathbb{Z}}^{n}$, a k-path with $l+1$ elements in ${\mathbb{Z}}^{n}$ is assumed to be a finite sequence ${({x}_{i})}_{i\in {[0,l]}_{\mathbb{Z}}}\subset {\mathbb{Z}}^{n}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=1$ [17].
- A digital image $(X,k)$ is said to be k-connected if, for any distinct points such as $x,y$ in $(X,k)$, there is a k-path ${({x}_{i})}_{i\in {[0,l]}_{\mathbb{Z}}}\subset X$ such that $x={x}_{0}$ and $y={x}_{l}$.
- For a digital image $(X,k)$, the k-component of $x\in X$ is defined to be the largest k-connected subset of $(X,k)$ containing the point x.
- We say that a simple k-path is a finite set ${({x}_{i})}_{i\in {[0,m]}_{\mathbb{Z}}}\subset {\mathbb{Z}}^{n}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if and only if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=1$ [17]. In the cases ${x}_{0}=x$ 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 ${\mathbb{Z}}^{n}$ [10], denoted by $S{C}_{k}^{n,l},l\ge 4,l\in {\mathbb{N}}_{0}\backslash \{2\}$, ${\mathbb{N}}_{0}$ is the set of even natural numbers [10,17] and is the finite set ${({x}_{i})}_{i\in {[0,l-1]}_{\mathbb{Z}}}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if and only if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=\pm 1(mod\phantom{\rule{0.166667em}{0ex}}l)$ [10].
- For a digital image $(X,k)$, a digital k-neighborhood of ${x}_{0}\in X$ with radius $\epsilon $ is defined in X as the following subset [10] of X:$${N}_{k}({x}_{0},\epsilon ):=\{x\in X\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{l}_{k}({x}_{0},x)\le \epsilon \}\cup \{{x}_{0}\},\phantom{\rule{2.em}{0ex}}$$$${N}_{k}(x,1)={N}_{k}(x)\cap X.\phantom{\rule{2.em}{0ex}}$$
- Rosenfeld [20] defined the notion of digital continuity of a map $f:(X,{k}_{0})\to (Y,{k}_{1})$ by saying that f maps every ${k}_{0}$-connected subset of $(X,{k}_{0})$ into a ${k}_{1}$-connected subset of $(Y,{k}_{1})$.

**Proposition**

**1**

**.**Let $(X,{k}_{0})$ and $(Y,{k}_{1})$ be digital images in ${\mathbb{Z}}^{{n}_{0}}$ and ${\mathbb{Z}}^{{n}_{1}}$, respectively. A function $f:(X,{k}_{0})\to (Y,{k}_{1})$ is (digitally) $({k}_{0},{k}_{1})$-continuous if and only if, for every $x\in X$, $f({N}_{{k}_{0}}(x,1))\subset {N}_{{k}_{1}}(f(x),1)$.

**Definition**

**1**

**.**Consider two digital images $(X,{k}_{0})$ and $(Y,{k}_{1})$ in ${\mathbb{Z}}^{{n}_{0}}$ and ${\mathbb{Z}}^{{n}_{1}}$, respectively. Then, a map $h:X\to Y$ is called a $({k}_{0},{k}_{1})$-isomorphism if h is a $({k}_{0},{k}_{1})$-continuous bijection, and further, ${h}^{-1}:Y\to X$ is $({k}_{1},{k}_{0})$-continuous. Then, we use the notation $X{\approx}_{({k}_{0},{k}_{1})}Y$. Moreover, in the case ${k}_{0}={k}_{1}:=k$, we use the notation $X{\approx}_{k}Y$.

**Definition**

**2**

**.**Let ${c}^{*}:=({x}_{0},{x}_{1},\cdots ,{x}_{n})$ be a closed k-curve in $({\mathbb{Z}}^{2},k,\overline{k},{c}^{*})$. A point x of $\overline{{c}^{*}}$, the complement of ${c}^{*}$ in ${\mathbb{Z}}^{2}$, is said to be interior to ${c}^{*}$ if it belongs to the bounded $\overline{k}$-connected component of $\overline{{c}^{*}}$.

- $(\star )$
- $MS{C}_{8}^{*}:=MS{C}_{8}\cup Int(MS{C}_{8})$ [4], where $MS{C}_{8}$ is a digital image 8-isomorphic to the digital image, i.e., $MS{C}_{8}:=S{C}_{8}^{2,6}:=\{{c}_{0}=(0,0),{c}_{1}=(1,1),{c}_{2}=(1,2),{c}_{3}=(0,3),{c}_{4}=(-1,2),{c}_{5}=(-1,1)\}$;
- $(\star )$
- $MS{C}_{4}^{*}:=MS{C}_{4}\cup Int(MS{C}_{4})$ [4], where $MS{C}_{4}$ is a digital image 4-isomorphic to the digital image, i.e., $MS{C}_{4}:=S{C}_{4}^{2,8}:=\{{v}_{0}=(0,0),{v}_{1}=(1,0),{v}_{2}=(2,0),{v}_{3}=(2,1),{v}_{4}=(2,2),{v}_{5}=(1,2),{v}_{6}=(0,2),{v}_{7}=(0,1)\}$; and
- $(\star )$
- $MS{C}_{8}^{\prime *}:=MS{C}_{8}^{\prime}\cup Int(MS{C}_{8}^{\prime})$ [4], where $MS{C}_{8}^{\prime}$ is a digital image 8-isomorphic to the digital image, i.e., $MS{C}_{8}^{\prime}:=S{C}_{8}^{2,4}:=\{{w}_{0}=(0,0),{w}_{1}=(1,1),{w}_{2}=(0,2),{w}_{3}=(-1,1)\}$.

**Definition**

**3**

**.**Let $((X,A),{k}_{0})$ and $(Y,{k}_{1})$ be a digital image pair and a digital image, respectively. Let $f,g:X\to Y$ be $({k}_{0},{k}_{1})$-continuous functions. Suppose there exist $m\in \mathbb{N}$ and a function $H:X\times {[0,m]}_{\mathbb{Z}}\to Y$ such that

- for all $x\in X,H(x,0)=f(x)$ and $H(x,m)=g(x)$;
- for all $x\in X$, the induced function ${H}_{x}:{[0,m]}_{\mathbb{Z}}\to Y$ given by ${H}_{x}(t)=H(x,t)$ for all $t\in {[0,m]}_{\mathbb{Z}}$ is $(2,{k}_{1})$-continuous;
- for all $t\in {[0,m]}_{\mathbb{Z}}$, the induced function ${H}_{t}:X\to Y$ given by ${H}_{t}(x)=H(x,t)$ for all $x\in X$ is $({k}_{0},{k}_{1})$-continuous. Then, we say that H is a $({k}_{0},{k}_{1})$-homotopy between f and g [24].
- 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. To be precise, ${H}_{t}(x)=f(x)=g(x)$ for all $x\in A$ and for all $t\in {[0,m]}_{\mathbb{Z}}$.Then, we call H a $({k}_{0},{k}_{1})$-homotopy relative to A between f and g and we say that f and g are $({k}_{0},{k}_{1})$-homotopic relative to A in Y, $f{\simeq}_{({k}_{0},{k}_{1})relA}g$ in symbols.

**Definition**

**4**

**.**For two digital images $(X,k)$ and $(Y,k)$ in ${\mathbb{Z}}^{n}$, if there are k-continuous maps $h:X\to Y$ and $l:Y\to X$ such that the composite $l\circ h$ is k-homotopic to ${1}_{X}$ and the composite $h\circ l$ is k-homotopic to ${1}_{Y}$, then the map $h:X\to Y$ is called a k-homotopy equivalence and is denoted by $X{\simeq}_{k\xb7h\xb7e}Y$. Moreover, we say that $(X,k)$ is k-homotopy equivalent to $(Y,k)$.

**Definition**

**5**

- In case X is pointed k-contractible, the k-fundamental group ${\pi}^{k}(X,{x}_{0})$ is trivial [24].

**Definition**

**6**

**.**Let $(X,k)$ be a digital image in ${\mathbb{Z}}^{3}$, and $\overline{X}:={\mathbb{Z}}^{3}\backslash X$. Then, X is called a closed k-surface if it satisfies the following:

- (1)
- In case $(k,\overline{k})\in \{(26,6),(6,26)\}$,
- (a)
- for each point $x\in X$, ${\left|X\right|}^{x}$ has exactly one k-component k-adjacent to x;
- (b)
- $|\overline{X}{|}^{x}$ has exactly two $\overline{k}$-components $\overline{k}$-adjacent to x; we denote by ${C}^{x\phantom{\rule{0.166667em}{0ex}}x}$ and ${D}^{x\phantom{\rule{0.166667em}{0ex}}x}$ these two components; and
- (c)
- for any point $y\in {N}_{k}(x)\cap X$ (or ${N}_{k}(x,1)$ in $(X,k)$), ${N}_{\overline{k}}(y)\cap {C}^{x\phantom{\rule{0.166667em}{0ex}}x}\ne \varphi $ and ${N}_{\overline{k}}(y)\cap {D}^{x\phantom{\rule{0.166667em}{0ex}}x}\ne \varphi $. Furthermore, if a closed k-surface X does not have a simple k-point, then X is called simple.

- (2)
- In case $(k,\overline{k})=(18,6)$,
- (a)
- X is k-connected,
- (b)
- for each point $x\in X$, ${\left|X\right|}^{x}$ is a generalized simple closed k-curve. Furthermore, if the image ${\left|X\right|}^{x}$ is a simple closed k-curve, then the closed k-surface X is called simple.

**Definition**

**7**

**.**In ${\mathbb{Z}}^{3}$, let ${S}_{{k}_{0}}$ (resp. ${S}_{{k}_{1}}$) be a closed ${k}_{0}$-(resp. a closed ${k}_{1}$-)surface, where ${k}_{0}={k}_{1}\in \{6,18,26\}$.

- Consider ${A}_{{k}_{0}}^{\prime}\subset {A}_{{k}_{0}}\subset {S}_{{k}_{0}}$ and take ${A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}\subset {S}_{{k}_{0}}$, where ${A}_{{k}_{0}}{\approx}_{({k}_{0},4)}MS{C}_{4}^{*}$, ${A}_{{k}_{0}}{\approx}_{({k}_{0},8)}MS{C}_{8}^{*}$, or ${A}_{{k}_{0}}{\approx}_{({k}_{0},8)}MS{C}_{8}^{\prime *}$ and, further, ${A}_{{k}_{0}}^{\prime}{\approx}_{({k}_{0},4)}Int(MS{C}_{4})$, ${A}_{{k}_{0}}^{\prime}{\approx}_{({k}_{0},8)}Int(MS{C}_{8})$, or ${A}_{{k}_{0}}^{\prime}{\approx}_{({k}_{0},8)}Int(MS{C}_{8}^{\prime})$, respectively.
- Let $f:{A}_{{k}_{0}}\to f({A}_{{k}_{0}})\subset {S}_{{k}_{1}}^{\prime}$ be a $({k}_{0},{k}_{1})$-isomorphism. Remove ${A}_{{k}_{0}}^{\prime}$ and $f({A}_{{k}_{0}}^{\prime})$ from ${S}_{{k}_{0}}$ and ${S}_{{k}_{1}}$, respectively.
- Identify ${A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}$ and $f({A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime})$ by using the $({k}_{0},{k}_{1})$-isomorphism f. Then, the quotient space ${S}_{{k}_{0}}^{\prime}\cup {S}_{{k}_{1}}^{\prime}/\sim $ is obtained by $i(x)\sim f(x)\in {S}_{{k}_{1}}^{\prime}$ for $x\in {A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}$ and is denoted by ${S}_{{k}_{0}}\u266f{S}_{{k}_{1}}$, where ${S}_{{k}_{0}}^{\prime}={S}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}$, ${S}_{{k}_{1}}^{\prime}={S}_{{k}_{1}}\backslash f({A}_{{k}_{0}}^{\prime})$, and the map $i:{A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}\to {S}_{{k}_{0}}^{\prime}$ is the inclusion map.

**Remark**

**1**

**.**In the quotient space ${S}_{{k}_{0}}\u266f{S}_{{k}_{1}}:={S}_{{k}_{0}}^{\prime}\cup {S}_{{k}_{1}}^{\prime}/\sim $, the subsets ${S}_{{k}_{0}}^{\prime}\backslash ({A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime})$ and ${S}_{{k}_{1}}^{\prime}\backslash f({A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime})$ in ${S}_{{k}_{0}}\u266f{S}_{{k}_{1}}$ are assumed to be disjoint and are not k-adjacent, where ${k}_{0}={k}_{1}:=k$. Then, the digital image $({S}_{{k}_{0}}\u266f{S}_{{k}_{1}},k)$ is called a (digital) connected sum of ${S}_{{k}_{0}}$ and ${S}_{{k}_{1}}$.

**Lemma**

**1.**

**Proof.**

- (1)
- for all $x\in MS{S}_{18}^{\prime},H(x,0)={1}_{MS{S}_{18}^{\prime}}$ as an identity map on the set $MS{S}_{18}^{\prime}$, say ${1}_{MS{S}_{18}^{\prime}}$, and $H(x,2)={C}_{\{{e}_{5}\}}$ as the constant map at the set $\{{e}_{5}\}$
- (2)
- for all $x\in MS{S}_{18}^{\prime}$, the induced function ${H}_{x}:{[0,2]}_{\mathbb{Z}}\to MS{S}_{18}^{\prime}$ given by ${H}_{x}(t)=H(x,t)$ for all $t\in {[0,2]}_{\mathbb{Z}}$ is $(2,18)$-continuous;
- (3)
- for all $t\in {[0,2]}_{\mathbb{Z}}$, the induced function ${H}_{t}:MS{S}_{18}^{\prime}\to MS{S}_{18}^{\prime}$ given by ${H}_{t}(x)=H(x,t)$ for all $x\in MS{S}_{18}^{\prime}$ is 18-continuous.Thus, we obtain H which is an 18-homotopy between ${1}_{MS{S}_{18}^{\prime}}$ and ${C}_{\{{e}_{5}\}}$.
- (4)
- Furthermore, for all $t\in {[0,2]}_{\mathbb{Z}}$, assume that the induced map ${H}_{t}$ on $\{{e}_{5}\}$ is a constant.

- Then, $MS{S}_{18}$ is indeed pointed 18-contractible (correction of the “non-18-contractibility” of $MS{S}_{18}$ in Theorem 4.3(3) of Reference [3] and Theorem 4.2(3) of Reference [4]). Moreover, it is proved to be a simple closed 18-surface (see Figure 1a) [3,4]. Using a method similar to the 18-homotopy of Equation (8), we observe that there is indeed an 18-homotopy relative to the set $\{{c}_{9}\}$ between ${1}_{MS{S}_{18}}$ and ${C}_{\{{c}_{9}\}}$, which is the constant map at $\{{c}_{9}\}$ (see Figure 2b),$$H:MS{S}_{18}\times {[0,3]}_{\mathbb{Z}}\to MS{S}_{18},\phantom{\rule{2.em}{0ex}}$$
- $MS{S}_{26}^{\prime}:=MS{S}_{18}^{\prime}$, which is 26-contractible [3,4] and is the minimal simple closed 26-surface (see Figure 1b). Finally, we obtain $(MS{S}_{26}^{\prime},26,6,{\mathbb{Z}}^{3})$ according to Equation (1). Moreover, the proof of the 26-contractibility of $MS{S}_{26}^{\prime}$ is trivially proceeded with the homotopy in Equation (8).

**Remark**

**2.**

- (1)
- the digital image $(T,6)$ is not a closed 6-surface.
- (2)
- $(T,6)$ is pointed 6-contractible.

**Proof.**

## 3. A Geometric Realization of a Simple Closed $\mathit{k}$-Surface

**Definition**

**8.**

**Remark**

**3.**

**Example**

**1.**

**Definition**

**9.**

**Example**

**2.**

- (1)
- Based on $MS{S}_{18}$, we observe that ${D}_{18}({c}_{1})$ is the set as the union of polygons formulated by the 18-cycles in ${M}_{18}({c}_{1})$, i.e., the union of the two triangles with boundary generated by the two 18-cycles $({c}_{0},{c}_{1},{c}_{9})$ and $({c}_{0},{c}_{1},{c}_{6})$ and the two rectangles with boundary formulated by the 18-cycles $({c}_{1},{c}_{2},{c}_{8},{c}_{9})$ and $({c}_{1},{c}_{2},{c}_{7},{c}_{6})$.
- (2)
- Based on $MS{S}_{18}^{\prime}$, we observe that ${D}_{18}({e}_{0})$ is the set which is the union of four triangles with boundary formulated by four 18-cycles in ${M}_{18}({e}_{0})$.
- (3)
- In terms of the methods used in Equations (1) and (2), based on $MS{S}_{6}$, we observe that ${D}_{6}({d}_{0})$ is the set as the union of twelve polygons (or regular rectangles) formulated by the twelve 6-cycles in ${M}_{6}({d}_{0})$.

**Definition**

**10.**

**Proposition**

**2.**

**Proof.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6**

**.**Unlike a typical surface (or a 2-dimensional topological manifold) in the Euclidean topological space $({\mathbb{R}}^{3},U)$, we observe that, given an ${S}_{k}$ and $x\in {S}_{k}$ motivated by the set, $|\phantom{\rule{0.166667em}{0ex}}{S}_{k}{\phantom{\rule{0.166667em}{0ex}}|}^{x}$, the sets ${M}_{k}(x)$ and ${D}_{k}(x),x\in {S}_{k}$ play important roles in establishing a geometric realization of the given ${S}_{k}$.

**Example**

**3.**

## 4. Euler Characteristics for Digital $\mathit{k}$-Surfaces and Connected Sums of Closed $\mathit{k}$-Surfaces

**Remark**

**7.**

**Proof.**

**Example**

**4.**

**Remark**

**8**

**.**Given an ${S}_{k}$ referred to in Example 4, Reference [14] considered only the simplicial complexes formulated by only 2-dimensional digital k-simplexes on ${S}_{k}$. Then, given an ${S}_{k}$ in ${\mathbb{Z}}^{3}$, it is obvious that it need not produce a polyhedron in ${\mathbb{R}}^{3}$. To be precise, according to the approach of Reference [14], since each of the sets

**Definition**

**11**

**.**For an ${S}_{k}$, the Euler characteristic of ${S}_{k}$ is defined by

**Remark**

**9**

**.**The approach using Definition 11 is consistent with the research of the Euler characteristic of a typical closed surface from algebraic topology and polyhedron geometry.

**Proposition**

**3.**

**Proof.**

**Example**

**5.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Example**

**6.**

## 5. The (Almost) Fixed Point Property for Digital $\mathit{k}$-Surfaces and Connected Sums of Closed $\mathit{k}$-Surfaces

- We say that a digital image $(X,k)$ in ${\mathbb{Z}}^{n}$ has the fixed point property (FPP) [30] if, for every k-continuous map $f:(X,k)\to (X,k)$, there is a point $x\in X$ such that $f(x)=x$.
- We say that a digital image $(X,k)$ in ${\mathbb{Z}}^{n}$ has the almost fixed point property (AFPP) [30,31] if, for every k-continuous self-map f of $(X,k)$, there is a point $x\in X$ such that $f(x)=x$ or $f(x)$ is k-adjacent to x. In general, we observe that the AFPP is a more generalized concept than the FPP.

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Remark**

**11.**

**Remark**

**12.**

## 6. Conclusions and a Further Work

- a development of a new type digital surface associated with a Khalimsky manifold.
- fixed point theory for many kinds of digital topological structures on ${\mathbb{Z}}^{n}$ in [33].
- given a typical surface X in pure topology and geometry, after developing a new type of $LF$-topological structure on X, $T(X)$, we can explore some connections related to Euler characteristics between X and $T(X)$.
- after improving the earlier digital homology groups [14] for digital images, we can propose some relationships between the current Euler characteristic and a certain invariant involving new homology groups for digital closed k-surfaces.

## Funding

## Conflicts of Interest

## References

- 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] - Han, S.-E. Connected sum of digital closed surfaces. Inf. Sci.
**2006**, 176, 332–348. [Google Scholar] [CrossRef] - 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 fundamental group and Euler characteristic of a connected sum of digital closed surfaces. Inf. Sci.
**2007**, 177, 3314–3326. [Google Scholar] [CrossRef] - Malgouyres, R.; Bertrand, G. A new local property of strong n-surfaces. Pattern Recognit. Lett.
**1999**, 20, 417–428. [Google Scholar] [CrossRef] - Malgouyres, R.; Lenoir, A. Topology preservation within digital surfaces. Graph. Model.
**2000**, 62, 71–84. [Google Scholar] [CrossRef][Green Version] - Malgouyres, R. Computing the Fundamental Group in Digital Spaces. IJPRAI
**2001**, 15, 1075–1088. [Google Scholar] [CrossRef][Green Version] - Rosenfeld, A.; Klette, R. Digital geometry. Inf. Sci.
**2002**, 148, 123–127. [Google Scholar] [CrossRef] - Han, S.-E. Non-product property of the digital fundamental group. Inf. Sci.
**2005**, 171, 73–91. [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] - Chen, L.; Cooley, D.H.; Zhang, J. The equivalence between definitions of digital images. Inf. Sci.
**1999**, 115, 201–220. [Google Scholar] [CrossRef] - Morgenthaler, D.G.; Rosenfeld, A. Surfaces in three dimensional digital images. Inf. Control.
**1981**, 51, 227–247. [Google Scholar] [CrossRef][Green Version] - Boxer, L.; Staecker, P.C. Fundamental groups and Euler characteristics of sphere-like digital images. Appl. Gen. Topol.
**2016**, 17, 139–158. [Google Scholar] [CrossRef] - Bykov, A.I.; Zerkalov, L.G.; Rodríguez Pineda, M.A. Index of a point of 3-D digital binary image and algorithm of computing its Euler characteristic. Pattern Recognit.
**1999**, 32, 845–850. [Google Scholar] [CrossRef] - Imiya, A.; Eckhardt, U. The Euler Characteristics of Discrete Objects and Discrete Quasi-Objects. Comput. Vis. Image Underst.
**1999**, 75, 307–318. [Google Scholar] [CrossRef] - Kong, T.Y.; Rosenfeld, A. Digital topology: Introduction and survey. Comput. Vis. Graph. Image Process.
**1989**, 48, 357–393. [Google Scholar] [CrossRef] - McAndrew, A.; Osborne, C. The Euler characteristic on the face-centred cubic lattice. Pattern Recognit. Lett.
**1997**, 18, 229–237. [Google Scholar] [CrossRef] - Saha, P.K.; Chaudhuri, B.B. A new approach to computing the Euler characteristic. Pattern Recognit.
**1995**, 28, 1955–1963. [Google Scholar] [CrossRef] - Rosenfeld, A. Digital topology. Am. Math. Mon.
**1979**, 86, 76–87. [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] - 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. 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. J. Math. Imaging Vis.
**1999**, 10, 51–62. [Google Scholar] [CrossRef] - 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.; Park, B.G. Digital Graph (k
_{0},k_{1})-Homotopy Equivalence and Its Applications. In Proceedings of the Conference on Topology and Its Applications, Washington, DC, USA, 9–12 July 2003. [Google Scholar] - Khalimsky, E. Motion, deformation, and homotopy in finite spaces. In Proceedings of the IEEE International Conferences on Systems, Man, and Cybernetics, Alexandria, VA, USA, 20 October 1987; pp. 227–234. [Google Scholar]
- Massey, W.S. Algebraic Topology; Springer: New York, NY, USA, 1977. [Google Scholar]
- Spanier, E.H. Algebraic Topology; McGraw-Hill Inc.: New York, NY, USA, 1966. [Google Scholar]
- Rosenfeld, A. Continuous functions on digital pictures. Pattern Recognit. Lett.
**1986**, 4, 177–184. [Google Scholar] [CrossRef] - Han, S.-E. Remarks on the preservation of the almost fixed point property involving several types of digitizations. Mathematics
**2019**, 7, 954. [Google Scholar] [CrossRef][Green Version] - Han, S.-E.; Yao, W. Euler characteristics for digital wedge sums and their applications. Topol. Methods Nonlinear Anal.
**2017**, 49, 183–203. [Google Scholar] [CrossRef] - Han, S.-E.; Jafari, S.; Kang, J.M. Topologies on ${\mathbb{Z}}^{n}$ which are not homeomorphic to the n-dimensional Khalimsky topological space. Mathematics
**2019**, 7. [Google Scholar] [CrossRef][Green Version] - Han, S.-E. Covering rough set structures for a locally finite covering approximation space. Inf. Sci.
**2019**, 480, 420–437. [Google Scholar] [CrossRef] - Han, S.-E. Marcus-Wyse topological rough sets and their applications. Int. J. Approx. Reason.
**2019**, 106, 214–227. [Google Scholar] [CrossRef] - Kang, J.M.; Han, S.-E.; Lee, S. The fixed point property of non-retractable topological spaces. Mathematics
**2019**, 7, 879. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Configuration of the pointed 18-contractibility of $MS{S}_{18}^{\prime}$ (

**a**) and $MS{S}_{18}$ (

**b**).

**Figure 3.**(

**a**) Configuration of the elements of ${M}_{18}({c}_{1})$ in $MS{S}_{18}$ for the point ${c}_{1}\in MS{S}_{18}$; (

**b**) explanation of the elements of ${M}_{18}({e}_{0})$ in $MS{S}_{18}^{\prime}$ for the point ${e}_{0}\in MS{S}_{18}^{\prime}$ (or $MS{S}_{26}^{\prime}$); and (

**c**) configuration of the elements of ${M}_{6}({d}_{0})$ for the point ${d}_{0}\in MS{S}_{6}$.

**Figure 4.**Explanations of the non-almost fixed point property (AFPP) of $MS{S}_{18}$ (

**a**) $MS{S}_{18}^{\prime}$ (or $MS{S}_{26}^{\prime}$) (

**b**).

© 2019 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 Theory for Digital *k*-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces. *Mathematics* **2019**, *7*, 1244.
https://doi.org/10.3390/math7121244

**AMA Style**

Han S-E. Fixed Point Theory for Digital *k*-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces. *Mathematics*. 2019; 7(12):1244.
https://doi.org/10.3390/math7121244

**Chicago/Turabian Style**

Han, Sang-Eon. 2019. "Fixed Point Theory for Digital *k*-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces" *Mathematics* 7, no. 12: 1244.
https://doi.org/10.3390/math7121244