    # Good Pairs of Adjacency Relations in Arbitrary Dimensions

In this text we show, that the notion of a "good pair" that was introduced in the paper "Digital Manifolds and the Theorem of Jordan-Brouwer" has actually known models. We will show, how to choose cubical adjacencies, the generalizations of the well known 4- and 8-neighborhood to arbitrary dimensions, in order to find good pairs. Furthermore, we give another proof for the well known fact that the Khalimsky-topology implies good pairs. The outcome is consistent with the known theory as presented by T.Y. Kong, A. Rosenfeld, G.T. Herman and M. Khachan et.al and gives new insights in higher dimensions.

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## 1 Introduction

In the text  the author has given a new framework to define -manifolds in together with a notion of “good pairs” of adjacency relations. Such a good pair makes it possible for a -manifold to satisfy a discrete analog of the Theorem of Jordan-Brouwer. This Theorem is a generalization of the Jordan-curve Theorem, which states that every simple closed curve in separates its complement in exactly two connected components and is itself the boundary of both of them. Brouwer showed that the statement is true for simple -manifolds in for all . It has been an open question since the beginnings of digital image analysis, if this is true in a discrete setting, so to speak in .

As the figure 1 shows, it is not even clear what a simple closed curve should look like in a discrete setting. And really, this depends on the adjacency we impose on the points of . We also see from the figure, that it is not enough to use only one adjacency for the base-set (background / white points) and the objects (foreground / black points), we have to use pairs of them. Unfortunately, not every pair of adjacencies is suitable because some even fail to make a ()-manifold out of the neighbors of a given point, and so they do not even satisfy the Theorem of Jordan-Brouwer. On these grounds the notion of a good pair arose and good pairs are the central topic of this article.

A solution for the points in the figure would be, to equip the black points with the 8-adjacency and use the 4-adjacency for the white ones. Then is clear that a discrete notion of the Jordan-theorem is true for this example. Figure 1: Depending on the adjacency relations we use for the black and the white points, respectively, the set of black points is connected (8-adjacency) or disconnected (4-adjacency). Also the set of white points may be connected (8-adjacency) or disconnected (4-adjacency). Only 4-adjacency is depicted.

For a long time adjacencies like the 4- and 8-neighborhood have been used, and of course, it is possible to generalize them to higher dimensions. This is done in this paper and we will see, which pairs of such relations give us good pairs. To do so, we will use the gridcube model of which is widely accepted and may be found in the book of A. Rosenfeld and R. Klette . It gives us a basic understanding of how these adjacencies may be build in high dimensions and once we have a good mathematical description for them, we may use it for the study of pairs of the adjacencies that we will call “cubical” because of the relation to this model.

In the 1980s E. Khalimsky  proposed a topological motivated approach with the so called Khalimsky-neighborhood. This topological notion gives also rise to graph-theoretic adjacencies and so it seems interesting to study it. Since it is already known, that these relations form good pairs, as seen in  and , we can use it as a test for the theory that also shows, how we are able to combine topological and graph-theoretic concepts.

The paper is organized as follows: We start with some basic definitions in section 2 where we do a tour through basic discrete topology and the graph-theoretic knowledge we use in this text, in section 3 the important concepts of the paper  are given and in section 4 we apply the theory to the aforementioned adjacency relations. We end the text with some conclusions in section 5.

## 2 Basic Definitions

### 2.1 Topological Basics

We use this section to introduce some basic topological notions. These stem from the usual set-theoretic topology as it might be found in any textbook on topology like the one of Stöcker and Zieschang , but we also introduce some facts given by P.S. Alexandrov in his text 111Actually, Paul Alexandroff is the same person as Pavel Sergeyevitch Alexandrov. The different names origin in a different transcription of the cyrillic letters in German and English..

###### Definition 1

A pair is called topological space for a set and set , the so called open sets or topology on , with the following properties:

A trivial topology on is the discrete topology . Please do not mistake the special “discrete” topology with the “discrete” setting we are working in. Even the may be equipped with a discrete topology and almost none of the discrete topologies we are referring to in this text are powersets of the base-set.

The subsets of , which have an open complement are called closed. An open set is called neighborhood of a point if is contained in .

A topological space that satisfies the following stronger claim instead of property (2), is called Alexandrov-space

All results for topological spaces are also true in Alexandrov-spaces. Topological spaces may be classified concerning the following separation properties:

###### Definition 2

A topological space may satisfy some of the separation axioms:

:

:

:

One can see, that every -space is also a -space. It is also true, that considering property (2’) interesing only for -spaces:

###### Lemma 1

An Alexandrov-space that satisfies the separation axioms or necessarily has the discrete topology.

#### Proof.

Let be a -space, and a neighborhood of . If , then we are done. Otherwise, there exists a in and by property , we may find a neighborhood , the contains but not . The intersection of all these sets is open and so, has to be discrete.

The proof for -spaces is analog.

To give a topology on a set , it is enough to give a certain family of open sets that can be used to generate all the open sets of by using set-theoretic union. This family is then called base of the topology . A topological space is called locally finite, if for any point in exists a finite open set and a finite closed set that both contain . In the following, we define how we can build new topological spaces from given ones.

###### Definition 3

Let , , be a family of topological spaces and let be their product and projections. The product topology is defined by the base

 B={⋂k∈Kp−1k(Ok):Ok∈Tk,K⊂I,K finite}.

The space is called topological product of the .

###### Definition 4

Let be a topological space and . With the topology

 T|A={O∩A:O∈T}

The set can be turned into a topological space . The topology is called subspace topology of with respect .

###### Definition 5

A mapping between two topological spaces , is called continuous, if for every the set is in .

###### Definition 6

A topological space is called connected, if it cannot be decomposed into two nonempty open sets:

 P=O1∪O2,O1,O2∈T,O1≠∅≠O2⇒O1∩O2≠∅.

A set is called connected, if it is connected in the subspace topology.

###### Lemma 2

Let be a connected topological space and let be a topological space. If is a continuous mapping, then is connected.

From the continuity of the projections in the definition of the product topology we can deduce the following

###### Lemma 3

A topological space is connected if and only if all of its factors are connected.

We define a path of length to be a continuous mapping . A path is closed if .

###### Definition 7

A topological space is called path-connected, if for any two points exists a path of length depending only on and , such that and .

The topological space is called locally path-connected if for every point and every neighborhood of a path-connected neighborhood exists.

###### Corollary 1

The following holds:

1. Path-connected spaces are connected.

2. Connected and locally path-connected spaces are path-connected.

###### Definition 8

Let and be topological spaces. A homotopy from to is a family of mappings , with the following property: The mapping , , is continuous. The set has the product topology.

Two functions are called homotopic, if a homotopy exists with and . If is constant then is called nullhomotopic.

A homotopy is called linear, if it is linear in .

Just like in the definition of paths, the set does not need to be the set in the discrete setting we are going to use arbitrary connected subsets of for instance with a fitting topology.

###### Definition 9

A topological space is called simply connected if any closed path is nullhomotopic.

This means that we continuously contract every closed path into one point.

###### Lemma 4

If is a union of two open simply connected subspaces with contractible intersection, then it is simply connected.

### 2.2 Alexandrov-Spaces

Every Alexandrov-space has an unique base that is given by the set of minimal neighborhoods of all points in the base-set. The minimal neighborhoods are easily identified as the intersections of all neighborhoods of a given point. Let be a point in an Alexandrov-space . We write to denote its minimal neighborhood. Analog we may find a minimal closed set containing a given point . We denote this set by . To create an analogy to the graph-theoretic background of most of this theory, we define

 {A}T(p):=({U}T(p)∪{C}T(p))∖{p} (1)

to be the adjacency of the point in . The set can be made to an Alexandrov-space in the subspace-topology.

Given a set we may analog define the sets:

 {U}T(M):={p∈P:⋁q∈Mp∈% {U}T‘OP(q)} (2)
 {C}T(M):={p∈P:⋁q∈Mp∈% {C}T(q)} (3)
###### Lemma 5

The set functions and are closure operators, they satisfy:

1. .

2. .

3. .

#### Proof.

The first property is trivial. To show the second one let . Therefore, it exists a such that . By the precondition we have and therefore .

To prove property 3, let , therefore, a exists in such that . If holds, then holds . Otherwise, a exists such that is in . By the property of an Alexandrov-space, the point has to be in and therefore in . The other inclusion follows from 2.

###### Lemma 6

Let be an Alexandrov-space that contains one point such that the only open neighborhood of is the set itself. Then is contractible.

#### Proof.

We define a homotopy by for and for each .

We show, that is continuos. Let be open.

Case 1: The point is in . W.l.o.g. . Therefore, the set is open.

Case 2: The point is not in . The the set is open.

###### Lemma 7

Let be an Alexandrov-space and , then the set is contractible. Therefore, the Alexandrov-space has a base of contractible open sets. In particular, the set is local contractible.

#### Proof.

We utilize Lemma 6 together with and .

It is possible to establish a notion of dimension in Alexandrov-spaces. It can also be found in Evako et.al. :

###### Definition 10

Let be a Alexandrov-space and .

• , if .

• , if there is a point in such that and for all exists a with .

• , if . The set has the subspace topology.

• If no exists such that then define .

###### Definition 11

We call a -surface, if has two points and is disconnected under .

The set is called -surface for , if is connected under and for all the set is a -surface.

A -surface is called -sphere, if is finite and it is simply connected for .

By Evako et.al. gilt:

###### Theorem 1

Let be a Alexandrov-space that is a -surface for . Then, for any point holds, that is simply connected.

###### Theorem 2

Every Alexandrov-space is a partial order and every partial order defines an Alexandrov-space.

### 2.3 The Khalimsky-Topology

In this section we study an important Alexandrov-topologies. To define it we start with a topologization of the set which we can interpret a a discrete line. What possibilities do we have to define a non-trivial topology on this set such that it is connected?

One can see, that the sets

 B={{x}:x∈Z,x≡0(2)}∪{{x−1,x,x+1}:x∈Z,x≡1(2)} (4)

and

 B′={{x}:x∈Z,x≡1(2)}∪{{x−1,x,x+1}:x∈Z,x≡0(2)} (5)

are bases of topologies. They differ only by a translation. Therefore, it seams reasonable to just choose one of them both. We will use the base and denote its generated topology by . Figure 2: Figure (a) shows a section of (Z,κ). The base of the topology is represented by ellipses. The base of the topology may also be depicted as a digraph. Figure (b) shows this.
###### Lemma 8

The Alexandrov-space is connected.

To go from here to the higher-dimensional case, we may view as a -fold topological product of . We denote the product topology with . By all we know so far, it is clear, that is connected. We call this class of spaces Khalimsky-spaces after E. Khalimsky .

###### Lemma 9

The Alexandrov-space is connected for all .

#### Proof.

This follows from Lemma 3.

###### Theorem 3

All Khalimsky-spaces , satisfy the separation theorem of Jordan-Brouwer.

#### Proof.

The proof is easy if one uses the methods of algebraic topology, because is isomorphic to a cell-decomposition of :

 Rn=({{i}:i∈Z,i≡0(2)}∪{(i−1,i+1):i∈Z,i≡1(2)})n (6)

The set denotes the open real interval between the integers and .

Since the Theorem of Jordan-Brouwer is true for any , , it has to hold for -dimensional Khalimsky-space.

We give another proof in section 4.3.

To establish structure on the points of the set we have to define some kind of connectivity relation. This might be done in terms of a (set-theoretic) topology as in the last section, or we may develop a graph-theoretic framework as in the following part of the text.

###### Definition 12

Given a set , a relation is called adjacency if it has the following properties:

1. is finitary: .

2. is connected under .

3. Every finite subset of has at most one infinite connected component as complement.

A set is called connected if for any two points in exist points and a positive integer such that , and for all . Compare this definition to the topological one we gave above.

The property 3 of an adjacency-relation is in for always satisfied.

In the text we will consider pairs of adjacencies on the set . In this pair represents the adjacency on a set , while represents the adjacency on .

Let be the set of all translations on the set . The generators of induce a adjacency in a natural way:

###### Definition 13

Two points of are called proto-adjacent, in terms , if there exists a such that or .

We can view the generators of a the standard base of .

Another important adjacency on is .

 ω(p):={q∈Z:|pi−qi|≤1,0≤i≤n} (7)

In the rest of the text let and be two adjacencies on such that for any holds

 π(p)⊂α(p),β(p)⊂ω(p). (8)
###### Lemma 10

The set is connected under .

## 3 Digital Manifolds

If we want to talk about -Manifolds in we have to give a proper definition. Unfortunatly, all the definitions known to the author from the literature are not usable in terms of generalization to higher dimension or for the unification of the topological and graph-theoretic approach. So it is necessary, to give a new definition that satisfies this two criteria. This is don in . The new definition is manly based on the so called separation property. It gives a description on how a discrete -manifold should look like locally.

### 3.1 The Separation Property

We call the set

 Ck={0,1}k×{0}n−k⊂Zn (9)

the -dimensional standard cube in . The set can be embedded in different ways in . A general -cube in is defined by a translation of a standard cube.

Indeed, we can construct any -cube from one point with generators in the following way:

 C={τe11⋅τekk(p):ei∈{0,1},i=1,…,k} (10)

The dimension of is then . We use this construction in the next definition.

###### Definition 14

Let , and be a -cube, . The complement of is in not separated by under the pair , if for every -component of and every -subcube of the following is true:

If is such that has maximal cardinality among all sets of this form, and the sets and are both nonempty and lie in one common -component of , then holds

 (τ1τ2)−1(τ1τ2(C∗)∩M′)⊂τ−11(τ1(C∗)∩M′)∩τ−12(τ2(C∗)∩M′). (11) Figure 3: C∗ has an intersection of maximal cardinality with the α-component M′. The sets τ1(C∗)∖M′ and τ2(C∗)∖M′ are nonempty and belong to a β-component of MC. Since τ1τ2(C∗)∩M′=∅, the property of definition 14 is satisfied for this C∗. But the set M′ separates MC in the cube C∗. Why?

In the following, we only consider the case when has at most one -component. This can be justified by viewing any other -component besides the one considered as part of the background, since there is no -connection anyway. This property also gets important if we study the construction of the simplicial complex.

A set has the separation property under a pair , if for every -cube , as in the definition 14 the set is in not separated by

The meaning of the separation property is depicted in the figure 4. Figure 4: The black points represent the set M in the given 3-cubes. The white points represent the complement of M. In the cases (a) to (c) the complement, which is connected, is separated by M. This separation is depicted by the gray plane spanned by C∗ and τ1τ2(C∗). In Figure (d) occurs no separation, since the only choice for C∗ would be a 1-cube, that contains only black points.
###### Definition 15

An -connected set , for , is a (digital) -manifold under the pair , if the following properties hold:

1. In any -cube the set is -connected.

2. For every the set has exactly two -components and .

3. For every and every the point is -adjacent to and .

4. has the separation property.

How should a -manifold look like globally in general? We do not know. But we might say, that a single point in might be considered as the inside of some object, i.e. that it might be separated by the other points. The way to do this is to require the set of neighbors of a point to be a -manifold. This justifies the following:

###### Definition 16

A pair of adjacency relations on is a separating pair if for all the set is a -manifold under .

### 3.2 Double Points

###### Definition 17

A point is a double point under the pair , if there exist points and and a simple222A translation is called simple if no other translation exists with , . translation with , and .

This concept is the key to a local characterization of the good pairs . Without it, one could not consistently define topological invariants like the Euler-characteristic. It means that an edge between points in a set can be crossed by an edge between points of its complement and these four points lie in a square defined by the corresponding adjacencies. This crossing can be seen as a double point, belonging both to the foreground and to the background. Also, mention the close relationship to the separation property, which is a more general concept of similar interpretation. For further insight, refer to the text . Figure 5: A double point p. The fat edges represent the α-adjacency. Only the relevant edges have been drawn for clarity. The dotted edge represents the β-adjacency of p and z. The black points are the β-neighbors of z.
###### Definition 18

A separating pair of adjacencies in is a good pair, if for every the set contains no double points.

## 4 Good Pairs of Adjacency Relations

We will study adjacencies in the sense of the gridcube-model. This is a common model in computer graphics literature and has nothing to do with the -cubes we talked about earlier. We use this model here to make it easy to study the adjacency relations in this section. For more on this topic refer to the Book of Rosenfeld and Klette 

We identify the points of with -dimensional unit-cubes with barycenters in the points of the lattice . The cube that represents the point can be expressed in euclidean space as . Those gridcubes may be interpreted as union of (polytopal) faces of different dimension. Any of its faces is again a gridcube, only with a lower dimension. Take, for instance, a 3-dimensional gridcube . It has, among others, the 0-dimensional face , the 1-dimensional face and the 2-dimenional face with the vertices and .

Two given gridcubes may share a -dimensional face for . This -face is just the intersection of both of them. So we might say that the elements and of intersect in a common vertex (0-face) with the coordinates . However, the elements and share a common -face.

In the rest of the text we will no longer make the gridcube model explicit. It just serves as an introduction to visualize the concepts that we use to analyze the discrete geometry even in higher dimensions333I find it a lot easier to imagine a four-dimensional cube, than a four-dimensional grid….

###### Definition 19

Two points are called -adjacent for , denoted by , if their corresponding gridcubes share a common -face. We call this adjacencies cubical.

Clearly, this kind of relation we just defined is an adjacency-relation in the sense of definition 12:

###### Lemma 11

The relation is an adjacency-relation on for every and all integers between and .

#### Proof.

First, we have to check that for any the set has only finite cardinality. It is easy to check, that is just as defined earlier and every for is a subset of . Since has Elements in , the relations must be finitary.

To see that is connected under any , , we observe that is just another interpretation for the relation defined earlier. Since is -connected as proven in  and every is a superset of , we conclude that is -connected.

The last property is in with trivially satisfied.

###### Lemma 12

The cubical adjacency may be represented in as the set:

 {(y1,…,yn)∈Zn:maxi=1,…,n{|xi−yi|}=1,1≤n∑i=1|xi−yi|≤n−k} (12)

#### Proof.

Let and be two points of such that . This means, the gridcubes corresponding to and share a common -face. Their distance in the maximum-metric may not be greater than 1. Furthermore, and may not share a single common -face for . That means, all of that -faces must be faces of common -faces. Therefore, the two points may not have more than coordinates in common.

###### Lemma 13

Let be a cubical adjacency on . It holds:

1. is invariant under translations

2. is invariant under permutations of coordinates.

#### Proof.

Let be any translation on . We need to show for any . From the representation of we may deduce:

 τ(α(p)) = τ({q∈Zn:q∈α(p)} (13) = {τ(q):q∈Zn,q∈α(p)} (14) = {q′∈Zn:q′∈α(τ(p))} (15) = α(τ(p)) (16)

The proof of the second part is analog.

What is the structure of the cubical adjacencies in ? We take a closer look at -dimensional cubes.

###### Lemma 14

The number of -faces of a -dimensional cube is

 (nk)⋅2n−k. (17)

#### Proof.

We use induction on the dimension of the cube.

For we observe, that a 0-dimensional cube is just a point and has only one 0-face. Therefore, the induction base is correct.

In the case , we notice that a -dimensional cube may be created from a -dimensional one by doubling the cube and inserting a -face for every -face in the original cube. Therefore, we get by induction hypothesis and Pascals Theorem:

 2(n−1k)⋅2n−1−k+(n−1k−1)⋅2n−1−(k−1) = [(n−1k)+(n−1k−1)]⋅2n−k (18) = (nk)⋅2n−k (19)

This proves the Lemma.

###### Lemma 15

For every , the number of -neighbors is

 |αk(p)|=n∑i=k(ni)⋅2n−i. (20)

#### Proof.

Obviously, any -face of a cube contains at least one -face for . Therefore, -adjacent cubes exist, that are also -adjacent. Since that are those, that share more than one common -face, the set for may be decomposed into the following disjoint sets:

 αk(p)={q∈Zn:p,q have at most one k-face in common }∪{q∈Zn:p,q have at most one (k+1)-face in common }⋮∪{q∈Zn:p,q have at most one (n−1)-face in common } (21)

By adding the cardinalities of these sets, which we can easily compute with the last Lemma we get the result . This proves the Lemma.

By this technique we get as examples of cubical adjacencies in the known 4- and 8-adjacencies, in the 6-, 18- and 26-adjacencies and in the 8-, 32-, 64- and 80-adjacencies.

### 4.2 Good Pairs of Cubical Adjacencies

In this section we will study, how we have to choose two cubical adjacencies to get to a good pair. We first will see, that it does not matter at which point of we study the adjacency, since the neighborhoods of all points look the same.

###### Lemma 16

Let be a cubical adjacency in . For any the set is graph-theoretical isomorphic to .

#### Proof.

This follows from the invariance under translations and the symmetry of the cubical adjacencies.

###### Lemma 17

Let be -connected. Then is also -connected for .

#### Proof.

Let be -connected. Thus, we have for any two a path such that for and for . By definition of , Lemma 12 and holds for and : and

 1≤n∑j=1|p(i)j−p(i−1)j|≤n−k

Therefore, we have for and the path is also a -path.

The next Lemmata help us understand, which adjacencies may be used as good pair.

###### Lemma 18

Let be a pair of cubical adjacencies on , . For any -cube as in section 3.1, the set is connected under if the following holds:

1. and , or

2. and .

#### Proof.

1. We use Lemma 17 and prove the proposition for

Let be any subcube of , that does not contain the point 0. We first show that is -connected. Suppose w.l.o.g. that the point is in and choose any other point . The point then has the form with

 maxi=1,…,n|ri|=1 and 1≤1+n∑i=2|ri|≤n−k. (23)

We select the smallest index such that and define

 r′=(1,r2,…,ri−1,0,ri+1,…,rn). (24)

The point is in :

 maxi=1,…,n|r′i|=1 and 1≤n∑i=1|r′i−ri|=|r′i−ri|=1≤n−(n−1). (25)

By iterating this process we get an -path from to .

Let now be and be two different -cubes. We may suppose w.l.o.g. that and . The two cubes contain a common point in since this point is -adjacent to and and it is in for :

 maxi=1,…,n|ti|=1 and n∑i=1|ti|=2≤n−k. (26)

Therefore, the set is -connected.

2. We show, that is connected under . By Lemma 17 this is enough.

The set contains all points , such that exactly one exists with and . Let and be two such points with . We have

 maxm=1,…,n|p(i)m−p(j)m|=1 and n∑m=1|p(i)m−p(j)m|=2≤n−(n−2). (27)

Therefore, and are -connected.

###### Corollary 2

Given a pair , then is -connected, if the following holds:

1. and or

2. and .

#### Proof.

This follows from the configuration of the -cubes in and the distribution of the -neighbors of 0 in those -cubes

###### Lemma 19

Let be a pair of cubical adjacencies on with . Then the set has exactly two -components for all .

#### Proof.

Obviously, 0 is in for any and it has no other -neighbors in .

We choose any point in . Then, contains points with . Those are not contained in in and form a -connected set. Therefore they are also -connected.

Define the set:

 ω(p)i:={s∈ω(p):|si|=2}. (28)

W.l.o.g. we consider that contains the point . It is easy to see, that either the point or the point is in .

Let be any point in . We construct a -path from to by defining the point

 s′=(2,…,p2,…,pi−1,pi,si+1,…,sn) (29)

with the smallest index such that . The point is a -neighbor of and after a finite number of iterations we have the -path from to . The sets and contain the points

 (p1,…,pi−1,2,pi+1,…,pn) (30)

and

 (p1,…,pj−1,2,pj+1,…,pn), (31)

respectively. In both sets the point

 t=(p1,…,pi−1,2,pi+1,…,pj−1,2,pj+1,…,pn) (32)

is contained and therefore, the sets are -connected.

It remains to show, that points in are -adjacent to one of the . Let , such that . In the case , then we have

 s′=(s0,…,si−1,si+1,si+1,…,sn)∈ω(p)i (33)

and in the case , it holds

 s′=(s0,…,si−1,si−1,si+1,…,sn)∈ω(p)i. (34)

Finally, we have to observe the case of the point with . Then, the point

 s′=(s1,…,si−1,pi,si+1…,sn) (35)

is a -neighbor of