Topological Complexities of Finite Digital Images
TTOPOLOGICAL COMPLEXITIES OF FINITE DIGITAL IMAGES
MELİH İS AND İSMET KARACA
Abstract.
Digital topological methods are often used on computing the topo-logical complexity of digital images. We give new results on the relation be-tween reducibility and digital contractibility in order to determine the topo-logical complexity of a digitally connected finite digital image. We present allpossible cases of the topological complexity TC of a finite digital image in Z and Z . Finally, we determine the higher topological complexity TC n of finiteirreducible digital images independently of the number of points for n > . Introduction
One of the main streams of topological robotics is to apply topological ideas tosolve specific problems of engineering and computer science. On the other hand,digital topology has an important place in the studies of computer science. Topo-logical robotics and digital topology have a common field of study and commonmethods. This raises the question: What results can one get in the subject ofrobotics by using topological methods on digital images? The answer gets inspiredwith the study of using discrete structures on computing topological complexitynumbers.Studies of topological robotics start with defining the notion of the topologicalcomplexity number of a path-connected topological space by Farber [14]. Thisnumber is an integer that indicates the complexity of area where the robot moves.Many different methods, especially cohomology, are used in algebraic topologyto determine the number exactly (see [15] for a collection of the methods used).Contractibility of a topological space is so important if one wants to know thetopological complexity number precisely. The topological complexity number of acontractible space is . If a topological complexity number of a topological spaceis , then the space must be contractible [14]. Rudyak [24] improves the idea ofthis topological complexity definition and presents the higher topological complex-ity number of a topological space. He proves that the special version of this newnumber corresponds to Farber’s topological complexity number. Karaca and Is [19]defines the digital topological complexity number and the digital higher topologicalcomplexity number [18] by moving the study to the field of digital topology. Digitaltopology is a discrete structure built on digital images at the point, so it assemblestopological features without including a topology (see [1–12], [20], and [21–23] for Date : September 2, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Topological complexity, higher topological complexity, digital topol-ogy, homotopy equivalence. a r X i v : . [ m a t h . GN ] S e p more information about digital topology, its some applications and digital geome-try). This fundamental difference makes some of the topological methods uselessin digital topology. For instance, cohomological cup-product method is one of thewell-known methods in usual topology to have a new bound for the topologicalcomplexity number [14]. But it does not work for digital images [18]. At this point,it is sometimes necessary to use new ways that comply with the rules of the digitaltopology. It is not only a problem of studies of digital topological complexity butalso a problem of studies in every aspect of digital topology. As an example, theEuler characteristic is not a homotopy invariant for digital images [13]. Staeckeret al. [16] have a new numerical homotopy invariant for digitally connected digitalimages and regard their invariant as ’true’, which means that it is not an adaptationfrom topology. They use the notions of reducibility and rigidity. In this paper, weexamine a relation between digital contractibility and reducibility (partly rigidity).This leads to us to have a characterization of finite digital images in Z and Z interms of the topological complexity and the higher topological complexity.First, we have a simple background of digital setting and recall the definitions ofthe topological complexity and the higher topological complexity with some impor-tant properties. Later, we show that if X is an irreducible digital image, then thetopological complexity of the image is greater than . We also demonstrate underwhat conditions the reducibility requires the digital contractibility. We prove thatif X ⊂ Z is a digitally connected finite image, then the topological complexity ofthe image is . After that, we examine the topological complexity of irreducibleimages having finite number of points. Using this fact, we have the topologicalcomplexity number of all digitally connected finite digital images in Z . We con-clude that there is no digitally connected finite image in Z and Z such that thetopological complexity of the image is greater than . In Section 4, we consider thediagonal map on a digital image X and define a new digital fibrational substituteof it. Then we find the digital higher topological complexity number of irreducibleimages with computing the digital Schwarz genus of the digital fibrational substi-tute. The topological complexity of the irreducible images is independent from thenumber of points. At the end of the paper, we state some open problems.2. Preliminaries
This section is planned to provide some backgrounds commonly used in digitaltopology and topological robotics.A digital image is the basic element of the digital topology and consists of aset with a relation on this set. More precisely, ( X, κ ) is a digital image [2], where X is a finite subset of Z n and κ is an adjacency relation for the points of X .On a digital image, unlike in topological spaces, there is an adjacency relationinstead of topology and this relation works as follows: Let X be a finite subsetof Z n and let k ∈ Z with ≤ k ≤ n . For any distinct elements x = ( x , ..., x n ) , y = ( y , ..., y n ) ∈ X , x and y are called c k − adjacent [2] if we have | x i − y i | = 1 for at most k indices i , and | x j − y j | (cid:54) = 1 implies x j = y j for all indices j . Thenotation x ↔ c k y is used when x is adjacent to y . By this construction, we have c = 2 adjacency in Z , c = 4 and c = 8 adjacencies in Z , and c = 6 , c = 18 and c = 26 adjacencies in Z . Let ( X, κ ) and ( Y, λ ) be any digital images. Let ( x , y ) and ( x , y ) be any two points in the cartesian product image X × Y . Then ( x , y ) and ( x , y ) are adjacent in X × Y [6] if one of the following conditions holds: • x = x and y = y ; or • x = x and y ↔ λ y ; or • x ↔ κ x and y = y ; or • x ↔ κ x and y ↔ λ y .Let ( X, κ ) be a digital image in Z n and let p be any point in X . A κ − neighbor [17]of p is the point that is κ − adjacent to p . Let ( X, κ ) ⊂ Z n be a digital image. X is called κ − connected [17] if and only if for every pair of different points x , y ∈ X , there is a set { x , x , ..., x m } of points in X such that x = x , y = x m and x i ↔ κ x i +1 for i = 0 , , ..., m − . Let f : ( X , κ ) → ( X , κ ) be a digital mapsuch that X ⊂ Z m and X ⊂ Z m . Then f is said to be ( κ , κ ) − continuous [2]if x ↔ κ x (cid:48) for any different x , x (cid:48) ∈ X , then f ( x ) ↔ κ f ( x (cid:48) ) in X . In addition, f is ( κ , κ ) − isomorphism [5] if f is bijective, ( κ , κ ) − continuous and the inverse f − is ( κ , κ ) − continuous.A set [ a, b ] Z = { z ∈ Z : a ≤ z ≤ b } is called a digital interval [4] from a to b . Sincethe interval is a subset of Z , it has − adjacency. If a digital map f : [0 , m ] Z → X is (2 , κ ) − continuous with f (0) = x and f ( m ) = y , then f is a digital path [4] from x to y in X . The digital path f is called a κ − loop if f (0) = f ( m ) . The product oftwo digital paths defined in [20]: Let f : [0 , m ] Z → X and g : [0 , n ] Z → X be digital κ − paths with f ( m ) = g (0) . Then the product of f and g is defined as the map ( f ∗ g ) : [0 , m + n ] Z → Xt (cid:55)−→ ( f ∗ g )( t ) = (cid:40) f ( t ) , t ∈ [0 , m ] Z g ( t − m ) , t ∈ [ m, m + n ] . Let ( X, κ ) and ( Y, λ ) be two digital images, and let f , g : X → Y be any ( κ, λ ) − continuous maps. The maps f and g are ( κ, λ ) − homotopic [2] if there exists m ∈ Z such that for all x ∈ X , there is a digital map F : X × [0 , m ] Z → Y with F ( x,
0) = f ( x ) and F ( x, m ) = g ( x ) , for any fixed t ∈ [0 , m ] Z , the digitalmap F t : X → Y is ( κ, λ ) − continuous and for any fixed x ∈ X , the digital map F x : [0 , m ] Z → Y is (2 , λ ) − continuous. It is denoted by f (cid:39) ( κ,λ ) g when f is ( κ, λ ) − homotopic to g . We also note that m is the step number of the homotopyin this construction. In other saying, we say that f is digitally homotopic to g in m step.Let f : X → Y be a ( κ, λ ) − continuous map. Then f is a ( κ, λ ) − homotopyequivalence [3] if there exists a ( λ, κ ) − continuous map g : Y → X for which g ◦ f isdigitally homotopic to the identity function on X and f ◦ g is digitally homotopicto the identity function on Y . A digital image X is said to be κ − contractible [2] ifthe identity map on X is ( κ, κ ) − homotopic to a constant map c for some x ∈ X ,where the constant map c : X −→ X is defined by c ( x ) = x for all x ∈ X . Definition 2.1. [16] Let ( X, κ ) be a finite digital image. If X is ( κ, κ ) − homotopyequivalent to an image of fewer points, then X is called reducible . If X is notreducible, then X is said to be irreducible . Definition 2.2. [16] Let ( X, κ ) be a finite digital image. If the identity map on X is the only map that is ( κ, κ ) − homotopic to the identity map on X , then X is rigid .Let ( X, κ ) be a digital image. If there is an integer m ≥ for which there existsa (2 , κ ) − continuous map f : [0 , m − Z → X such that the following conditionshold: • f is bijective; • f (0) ↔ κ f ( m − ; and • for all t ∈ [0 , m − Z , the only κ − neighbors of f ( t ) in f ([0 , m − Z ) are f (( t − mod m ) and f (( t + 1) mod m ) ,then X is a digital simple closed κ − curve [3]. A simple closed curve with m pointsis generally denoted by C m and named as an m − gon or a digital m − cycle . Let( X, κ ) be a digital image. An m − loop [16] is a digitally continuous map from C m to X . Moreover, the map p is called a simple m − loop if p is an injection with p ( c i ) ↔ κ p ( c i +1 ) in X such that there are no other adjacencies between points inthe image of C m . Proposition 2.3. [16] C m is irreducible for m ≥ . Definition 2.4. [16] L m ( X ) is an integer which counts the number of equivalenceclasses of m − loops for any finite digital image X . Theorem 2.5. [16] Let ( X, κ ) and ( Y, λ ) be any two digital images such thatthey are digitally homotopy equivalent. Then for all positive integer m , we get L m ( X ) = L m ( Y ) . The next three results are the basic facts that we often use in next sections. Byusing these results, we have an idea about the digital topological complexity of afinite digital image (reducible or irreducible) with respect to the number of points.
Proposition 2.6. [16] Let ( X, κ ) be a finite digital image. If X has no simple m − loop for any m ≥ , then X is digitally homotopy equivalent to a one-pointdigital image. Proposition 2.7. [16] Let ( X, κ ) be a digitally connected digital image having m points. If m ≤ , then X is digitally homotopy equivalent to a one-point digitalimage. Proposition 2.8. [16] Let X be a digitally connected digital image having fivepoints. Then X is digitally homotopy equivalent to a one-point digital image or to C . Let
P X be a set of all digitally continuous digital paths for any κ − connecteddigital image ( X, κ ) . Let s : X × X → P X be the digital map which takes any pair ( a, b ) of a digital image to a digital path starting at a and ending at b , is denoted bythe digital version of motion planning algorithm. In [19], there is a reasoned wayto define the continuity of motion planning algorithm. The digital connectednesson P X is defined as follows: let τ be an adjacency relation on P X , and let α and β be any digital paths on X . If α and β are τ − connected for all t ∈ [0 , m ] Z , then α ↔ κ β . α and β can have different steps in their way. For instance, when α has steps and β has steps, the last step of β repeats itself times. Then both α and β have the same number of steps, which means there is no confusion about theadjacency of digital paths. See [19] for more detail and example about continuityof digital motion planning algorithm. Moreover, π : P X → X × X is a digital map,which takes any digital path α to the pair ( α (0) , α ( m )) , where α ( m ) is the finalstep of α . Finally, we are ready to give the definition: Definition 2.9. [19] The digital topological complexity TC ( X, κ ) is the minimalnumber k such that X × X = U ∪ U ∪ ... ∪ U k with the property that there exists a digitally continuous motion planning algorithm s j : U j → P X , j = 1 , , ..., k , for which π ◦ s j is identity map over each U j ⊂ X × X .If no such k exists, then TC ( X, κ ) = ∞ .We compute the digital topological complexity of only connected digital images(recall that in ordinary topology, only path-connected topological spaces are con-sidered for the computation of the topological complexity). The next propositionis quite important such as the fact that the topological complexity is a homotopyinvariant. Proposition 2.10. [19] TC ( X, κ ) = 1 if and only if ( X, κ ) is κ − contractible. Definition 2.11. [18] Let f : ( X, κ ) → ( Y, λ ) be a map in digital images withdigitally connected spaces ( X, κ ) and ( Y, λ ) . A digital fibrational substitute of f is defined as a digital fibration (cid:98) f : ( Z, κ ) −→ ( Y, κ ) such that there exists acommutative diagram X h (cid:47) (cid:47) f (cid:15) (cid:15) Z (cid:98) f (cid:15) (cid:15) Y Y Y, where h is a digital homotopy equivalence.Let p : X → Y be a digital fibration. The digital Schwarz genus [18] of p isdefined as the minimum number k such that X = U ∪ U ∪ ... ∪ U k with theproperty that for all ≤ i ≤ k , there is a digitally continuous map s i : U i → X that satisfies p ◦ s i = id U i . If we do not have a digital fibration, then we regard thedigital Schwarz genus of a map as the digital Schwarz genus of its digital fibrationalsubstitute. Consequently, we now give another important definition: Definition 2.12. [18] Let X be any κ -connected digital image. Let J n be thewedge of n − digital intervals [0 , m ] Z , ..., [0 , m n ] Z for a positive integer n , where i ∈ [0 , m i ] , i = 1 , ..., n , are identified. Then the digital higher topological complexity TC n ( X, κ ) is defined by the digital Schwarz genus of the digital fibration e n : X J n → X n f (cid:55)−→ ( f ( m ) , ..., f ( m n ) n ) ,where ( m i ) k , k = 1 , ..., n denotes the endpoints of the i − th interval for each i . In the definition of the higher topological complexity in digital images, we haveTC = TC [18]. Furthermore, TC n is also a homotopy invariant for digital imagesjust as TC. 3. Digital Topological Complexity in Z and Z We begin with discussing the relation between the contractibility and the re-ducibility on digitally connected digital images. It is clear that if ( X, κ ) is a κ − connected and κ − contractible finite digital image, then X is reducible. Theconverse need not to be true. For example, consider the following digital image X with − adjacency and its digital homotopy equivalence in Figure 3.1: Figure 3.1.
The digital image X with − adjacency is on the left(a) and its digital homotopy equivalence X \ { (3 , − } on the right(b).The digital image X is reducible because it is digitally homotopy equivalent tothe image X \ { (3 , − } (Figure 3.1 (b)) but it is well-known that X is not − contractible. Combining this result with Proposition 2.10, we have that thetopological complexity number of a reducible image can be different from . In-deed, we obtain thatTC ( X,
8) = TC ( X \ { (3 , − } ,
8) = 2 (see [Example 3.5, [19]]). In addition, if for any digitally connected finite image ( X, κ ) having more than one point, TC ( X, κ ) = 1 , then X must be reducible. So,we immediately have the result: Proposition 3.1.
Let ( X, κ ) be a digitally connected finite image having more thanone point. If X is irreducible, then TC ( X, κ ) > . We note that Proposition 3.1 is still true if we choose X as a rigid digital imageinstead of an irreducible digital image. We express that the digital contractibilityimplies the reducibility. The next Lemma shows that the converse of this expressionis valid. Lemma 3.2.
Let X be a digital image with m ≥ points. a) If C m is an empty set, then X is digitally contractible if and only if X isreducible. b) If C m is nonempty and X is not digitally homotopy equivalent to C m , then X is digitally contractible if and only if X is reducible.Proof. a) It is enough to prove that if X is reducible then X is digitally contractible.Let X be a reducible digital image. Then X is digitally homotopy equivalent to animage X \ A , where A has fewer points than X . Let ∗ be any point of X . If X isdigitally homotopy equivalent to the one-point image {∗} , then there is nothing toprove. Assume that X is not digitally homotopy equivalent to the one-point image.By Proposition 2.6, we have that X has a simple m − loop for any m ≥ . Therefore,there exists a digitally continuous injection p : C m → X . This is a contradictionbecause p cannot be an injection. Whereas X has m points, C m is empty for any m ≥ . As a conclusion, X is digitally contractible. b) Let X be a reducible digital image. Assume that X is not digitally homotopyequivalent to the one point image. Then we have a digitally continuous injection p : C m → X . The cardinality of C m and m and the cardinality of X is thesame. This implies that p is surjective. Therefore, p is a bijection. If we define q : X → C m with q ( x ) = p − ( x ) , then q is digitally continuous. Indeed, for any x i ∈ X , i = 1 , ..., m , we find p − ( x i ) ↔ p − ( x i +1 ) because p − ( x i ) = c i and p − ( x i +1 ) = c i +1 . Hence, we get p ◦ q = id X and q ◦ p = id X . This means that X isdigitally homotopy equivalent to C m which is a contradiction. Finally, X is digitallyhomotopy equivalent to the one point image, i.e. X is digitally contractible. (cid:3) Lemma 3.3.
A digitally connected image X ⊂ Z is − contractible if and only if L ( X ) = 1 .Proof. Let X ⊂ Z be a − contractible image. Then X is digitally homotopyequivalent to the one-point digital image {∗} . We observe that the one-point is theunique irreducible image in Z . By Theorem 2.5, L ( X ) = L ( {∗} ) = 1 . Conversely,if L ( X ) = 1 , then we have that the number of equivalence classes of − loops is .This means that X is − contractible. (cid:3) From the digital image X in Figure 3.1 (a), we cannot generalize Lemma 3.3 in Z n for n > . Since X is − connected, L ( X ) = 1 . However, X is not − contractible.The following Corollary is a result of Lemma 3.3 and Proposition 2.10. Corollary 3.4.
Let X ⊂ Z be a digitally connected finite image. Then we getTC ( X,
2) = 1 . We now provide the digital topological complexity numbers of digital simpleclosed curves in Z . Theorem 3.5.
Let C m be a nonempty κ − connected digital simple closed curve forany positive integer m , where κ ∈ { , } . ThenTC ( C m , κ ) = (cid:40) , m < , m > . Proof.
There are two adjacency relations and in Z so we have two cases.First, consider the − adjacency on C m . We catalog the first nonempty simpleclosed curves with respect to the number m in this case (see Figure 3.2). We Figure 3.2.
Nonempty simple closed curve C m related to − adjacency for m = 1 , ..., .note that some graphics can be different (but homotopy equivalent) in Figure 3.2.For instance, the points of C can be drawn vertically. This does not effect theresult as the digital topological complexity number is a homotopy invariant fordigital images. For m > , the list is extended. However, the computation of TCchanges only when m > . Let m < . We have TC ( C m ,
4) = 1 because theyare − contractible digital images. If m > , then we show that TC ( C m ,
4) = 2 .Let us choose any two diagonally points (the diagonal can be from left to rightor from right to left) on any squares or rectangles for any m > and dividethe graphic into two parts named as U and U . Without loss of generality, weassume that U has one of the diagonal points and U has the other point. Then U and U have the same number of points. We set V = { ( x, y ) ∈ C m × C m | ( x, y ) ∈ U } and V = { ( x, y ) ∈ C m × C m | ( x, y ) ∈ U or x ∈ U , y ∈ U or x ∈ U , y ∈ U } as the subsets of C m × C m . Therefore, we get C m × C m = V ∪ V . In addition,there exist digitally continuous sections s : V → P C m and s : V → P C m ofa digital fibration π : P C m → C m × C m . These satisfy that π ◦ s = id V and π ◦ s = id V and give the desired result for − adjacency. Similarly, we list the first nonempty simple closed curves with − adjacency in Figure 3. For m < , C m is − contractible. Then we have that TC ( C m ,
8) = 1 . For m > , we choose thetop and the bottom point of C m (if there are one more top or bottom points, thenchoose one pair of them such that they are located vertically according to the eachother) and divide the graphic into two parts named as T and T . Without loss ofgenerality, we assume that T has the bottom point and T has the top point. Weset W = { ( x, y ) ∈ C m × C m | ( x, y ) ∈ T } and W = { ( x, y ) ∈ C m × C m | ( x, y ) ∈ T or x ∈ T , y ∈ T or x ∈ T , y ∈ T } as the subsets of C m × C m . Then we have digitally continuous sections t : W → P C m and t : W → P C m of a digital map π : P C m → C m × C m Figure 3.3.
Nonempty simple closed curve C m related to − adjacency for m = 1 , ..., .that satisfy that the digital maps π ◦ t and π ◦ t equal to the identity maps.Moreover, C is an empty set for both and adjacencies. This completes theproof. (cid:3) Corollary 3.6.
Let X ⊂ Z be a digitally connected digital image with m points.TC ( X,
8) = 1 for m < and TC ( X,
4) = 1 for m < .Proof. Let m ≤ . By Proposition 2.7, X is digitally homotopy equivalent to theone-point digital image. Then, we have that TC ( X, κ ) = 1 , where κ ∈ { , } .Let m = 5 . From Proposition 2.8 and Proposition 2.10, we get TC ( X, κ ) = 1 ,where κ ∈ { , } . Let m = 6 or . Then C m is an empty set with respect to − adjacency. Then X is digitally contractible because X is reducible. This showsthat TC ( X,
4) = 1 for m = 6 or m = 7 . (cid:3) We are now ready to compute the topological complexity number of any finitedigital image in Z . This characterization indicates that there is no any finite digitalimage in Z whose topopological complexity number is greater than 2. Corollary 3.7.
Let X ⊂ Z be a κ − connected digital image with m points. If C m (cid:54) = ∅ and X is digitally homotopy equivalent to C m , then we get thatTC ( X, κ ) = (cid:40) , κ = 4 and m < , κ = 8 and m < and TC ( X, κ ) = (cid:40) , κ = 4 and m ≥ , κ = 8 and m ≥ . Otherwise, we have that TC ( X, κ ) = 1 , where κ ∈ { , } for any m .Proof. By Theorem 3.5 and Corollary 3.6, it is enough to show thatTC ( X,
4) = TC ( X,
8) = 1 when C m = ∅ or X is not digitally homotopy equivalent to C m . Let C m = ∅ . If m ≤ , then the result holds from Proposition 2.7. If m ≥ , then we have that X is reducible from Proposition 2.3. Hence, the first part a) of Lemma 3.2 givesthe desired result. Assume that the digital image X is not digitally homotopy equivalent to C m . Then TC ( X, κ ) (cid:54) = 2 . Let C m be nonempty and let m ≥ . Since C m is irreducible for m ≥ , X is reducible. Thus, the second part b) of Lemma3.2 completes the proof. (cid:3) Digital Higher Topological Complexity of Finite D DigitalImages
We aim to give a general characterization for the digital higher topological com-plexity computations of any finite digital image especially in Z in this section.We begin with computing the digital higher topological complexity TC n of anyone-point digital image for n ≥ . Consider X = {∗} ⊂ Z with − adjacency.Let f ∈ X J n be a constant map at ∗ . The digital fibration e n : X J n → X n ,defined by e n ( f ) = ( ∗ , ∗ , ..., ∗ ) , has a digitally continuous map s : X n → X J n with s ( ∗ , ∗ , ..., ∗ ) = f such that e n ◦ s = id . This shows that TC n ( X ) = 1 , where X is aone-point digital image. Theorem 4.1.
Let ( X, κ ) be a finite κ − connected digital image in Z and n ≥ bean integer. Then TC n ( X, κ ) = 1 .Proof. If ( X, κ ) is finite and κ − connected in Z , then it is easy to see that X is κ − contractible. Hence, it is κ − homotopy equivalent to the one-point digital image.The digital homotopy invariance of TC n gives the desired result. (cid:3) The digital higher topological complexity computation of a one-point digital im-age is quite useful because a great majority of digital images in Z is digitallycontractible (have the same homotopy type with the one-point image). We nowexamine the digital higher topological complexity of another type which is nothomotopy equivalent to the one-point image. Lemma 4.2.
Let ( X, κ ) be a κ − connected digital image. Consider the set S n ( X ) = { ( f, p , p , ..., p n ) | p i ∈ Im ( f ) , f is a digital path in X, i = 1 , , ..., n } in X [0 ,m ] Z × X n . Then the digital map e (cid:48) n : S n ( X ) −→ X n ( f, p , p , ..., p n ) (cid:55)−→ ( p , p , ..., p n ) is a digital fibrational substitute of the diagonal map d n : X → X n . Remark 4.3.
Note that the adjacency relation on S n ( X ) is defined as follows:for all ( f, p , p , ..., p n ) , ( g, q , q , ..., q n ) ∈ S n ( X ) , ( f, p , p , ..., p n ) is κ ∗ − adjacentto ( g, q , q , ..., q n ) if f is λ − adjacent to g and p i is κ − adjacent to q i for all i = 1 , , ..., n , where κ ∗ is an adjacency relation on X [0 ,m ] Z × X n and λ is anadjacency relation on digital paths in X .Proof. Let d n : X → X n be a diagonal map of X . Define the digital map h : X → S n ( X ) by h ( x ) = ( (cid:15) x , x, x, ..., x ) , where (cid:15) x is the digital constant pathat x . Let ( f, p , p , ..., p n ) ∈ S n ( X ) . Then there exists y ∈ X such that f (0) = y .Since X is κ − connected, there exists a digital path g from x to y in X , i.e. g (0) = x and g (1) = f (0) = y . To show that h is a digital homotopy equivalence, we define a digital map k : S n ( X ) → X with k ( f, p , p , ..., p n ) = f ∗ g (0) . It is easy to seethat h ◦ k is digitally homotopic to identity map on S n ( X ) and k ◦ h is digitallyhomotopic to identity map on X . Moreover, we find e (cid:48) n ◦ h ( x ) = e (cid:48) n ( (cid:15) x , x, ..., x ) = ( x, x, ..., x ) = d n ( x ) . Consequently, e (cid:48) n is a digital fibrational substitute of d n . (cid:3) Lemma 4.4. TC ( C ,
8) = 2 .Proof.
Let X = C = { p = (0 , , p = (1 , , p = (2 , , p = (3 , ,p = (2 , − , p = (1 , − } , where p < p < p < p < p < p (see Figure 4.1). Let e (cid:48) : S ( X ) → X be adigital fibration with e (cid:48) ( f, p i , p j , p k ) = ( p i , p j , p k ) for i, j, k ∈ { , , , , , } . Wedivide X into two parts. A consists of triples in C such that the order of pointsnever changes from left to right, i.e. p i ≤ p j ≤ p k or if p i > p j , then p i = 6 and p j = 1 (similarly if p j > p k , then p j = 6 and p k = 1 ). A consists of elements of C in which they do not belong to A , i.e. the order of points can change fromleft to right except using and consecutively. Let ( p i , p j , p k ) ∈ A . Using thesepoints, we set a route starting and ending at p i and p k , respectively. Then we havea digitally continuous map s : A → S ( X ) with s ( p i , p j , p k ) = ( f, p i , p j , p k ) ,where f is the route (digital path from p i to p k ). It is clear that e (cid:48) ◦ s = id S ( X ) .Similarly, we can construct s : A → S ( X ) with s ( p i , p j , p k ) = ( f, p i , p j , p k ) over A . Hence, we find that e (cid:48) ◦ s = id S ( X ) . Moreover, we have that X = A ∪ A .As a result, we get genus κ ∗ ,λ ∗ ( e (cid:48) ) = 2 , where κ ∗ and λ ∗ are adjacency relations on S ( X ) and X , respectively. (cid:3) Figure 4.1. C with the order of points in it Lemma 4.5. TC ( C ,
4) = 2 .Proof.
Let X = C = { r = (0 , , r = (0 , , r = (0 , , r = (1 , , r = (2 , ,r = (2 , , r = (2 , , r = (1 , } , where r < r < r < r < r < r < r < r (see Figure 4.2). In a similar way ofLemma 4.4, we get B without changing the order of points and t : B → S ( X ) isa digitally continuous map over C such that e (cid:48) ◦ t is identity over B . Changingthe order of points in C , we set B that consists of triples in C × C × C . Thedigitally continuous map t : B → S ( X ) gives us e (cid:48) ◦ t is identity over S ( X ) .Hence, we divide X into two parts B and B . This proves that TC ( X,
4) = 2 . (cid:3) Figure 4.2. C with the order of points in it Corollary 4.6.
Let C m be a nonempty and κ − connected digital simple closed curve.Then TC ( C m ,
4) = 2 for m ≥ and TC ( C m ,
8) = 2 for m ≥ .Proof. The proof is a generalization of Lemma 4.4 and Lemma 4.5. The order ofpoints in C m can be easily constructed for all cases. (cid:3) Corollary 4.6 can be improved for n > and TC n gives the same result with TC for irreducible digital images: Theorem 4.7.
Let C m be a nonempty and κ − connected digital simple closed curveand n > be a positive integer. Then • TC n ( C m ,
4) = 2 , for m ≥ , • TC n ( C m ,
8) = 2 , for m ≥ .Proof. Let m ≥ n . Let p , ..., p m be points of C m , where p < p < ... < p m . Byusing the order, a digital path can be obtained by taking n or less (staying on thesame point more than once) of m points. Then the method of Lemma 4.4 works forthis case. Let m < n and ( f, p , p , ..., p n ) ∈ S n ( X ) . In this case, it is necessary toincrease the number of steps of the digital path to be able to have an n − step path created with m points. A new n − step path is obtained by adding the endpoint ofany m − step path f to the end of the path m − n times. Since we have n − steppath, we use its n points in the definition of S n ( X ) . After that, we divide X n intotwo parts A and A again: n points of the digital image in which following theorder and not, respectively. Thus, we conclude that the digital Schwarz genus of e (cid:48) n is . (cid:3) Conclusion
The aim of this paper is to characterize the digital topological complexity ofdigitally connected two dimensional finite digital images entirely. We first dealwith simple closed curves among digital images because they are irreducible. Aftergiving the results about digital simple closed curves, we examine the topologicalcomplexity and the higher topological complexity of all possible digitally connectedfinite digital images in Z and Z .One of the open problems on this topic is to apply our works on − dimensionaldigital images. As the number of points that a digital image has in three-dimensional space extremely increases, it is not easy to categorize the topologicalcomplexities of these points. Before solving this problem, it is more convenient thattrying to categorize the digital images up to digital homotopy equivalence, becauseof the fact that the topological complexity (and the higher topological complex-ity) is a homotopy invariant for digital images. Moreover, one can observe theresults about the topological complexities of reducible or irreducible images in Z .This leads us to think more about the characterize digital images up to the digitalhomotopy equivalence in any dimension of digital topology. Acknowledgment.
The first author is granted as fellowship by the Scientificand Technological Research Council of Turkey TUBITAK-2211-A. In addition,this work was partially supported by Research Fund of the Ege University(Project Number: FDK-2020-21123)
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E-mail address : [email protected] Ismet Karaca Ege University, Faculty of Science, Department of Mathematics, Izmir,Turkey
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