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Galois connections between sets of paths and closure operators in simple graphs

Josef Šlapal
• Corresponding author
• IT4Innovations Centre of Excellence, Brno University of Technology, 612 66 Brno, Czech Republic
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Published Online: 2018-12-31 | DOI: https://doi.org/10.1515/math-2018-0128

Abstract

For every positive integer n,we introduce and discuss an isotone Galois connection between the sets of paths of lengths n in a simple graph and the closure operators on the (vertex set of the) graph. We consider certain sets of paths in a particular graph on the digital line Z and study the closure operators associated, in the Galois connection discussed, with these sets of paths. We also focus on the closure operators on the digital plane Z2 associated with a special product of the sets of paths considered and show that these closure operators may be used as background structures on the plane for the study of digital images.

MSC 2010: 05C10; 54A05; 54D05

Introduction

It is useful to find new relationships between different mathematical structures or theories because they demonstrate the interconnectedness of mathematics and enable us to use tools of one theory to solve problems of another. In the present paper, we will discuss certain relationships between graph theory and topology. We will introduce and study Galois connections between the sets of paths of the same length in a graph and closure operators on the vertex set of the graph. We will focus on the connectedness provided by the closure operators associated, in the Galois connections introduced, to certain sets of paths in the 2-adjacency graph on the digital line Z. The closure operators will be shown to generalize the connected ordered topologies [1], hence the Khalimsky topology, on Z. Further, we will discuss the closure operators on the digital plane Z2 associated with special products of the sets of paths with the same length in the 2-adjacency graph on Z. These closure operators include the Khalimsky topology on Z2 and we will show that they allow for a digital analogue of the Jordan curve theorem (recall that the classical Jordan curve theorem states that a simple closed curve in the real (Euclidean) plane separates the plane into precisely two components). It follows that the closure operators may be used as background structures on the digital plane Z2 for the study of digital images.

The idea of studying connectedness in a graph with respect to sets of paths is taken from [2]where certain special sets of paths, called path partitions, were used to obtain connectedness with a convenient geometric behavior. In the present note, we will employ arbitrary sets of paths (of given lengths) and investigate connectedness with respect to the closure operators associated with the sets in a Galois connection.

For the graph-theoretic terminology, we refer to [3]. By a graph G = (V, E), we understand an (undirected simple) graph (without loops) with VØ as the vertex set and E ⊆ {{x, y}; x, y ∈ V, xy} as the set of edges. We will say that G is a graph on V. Two vertices x, y 2 V are said to be adjacent (to each other) if {x, y} ∈ E. Recall that a path in G is a (finite) sequence (xi| in), i.e., (x0, x1, ..., xn), of pairwise different vertices of V such that xi is adjacent to xi+1 whenever i < n. The non-negative integer n is called the length of the path (xi| in). A sequence (xi| in) of vertices of G is called a circle if n > 2, x0 = xn, and (xi| i < n) is a path. Given graphs G1 = (V1, E1, ) and G2 = (V2, E2), we say that G1 is a subgraph of G2 if V1 ⊆ V2 and E1 ⊆ E 2. If, moreover, V1 = V2, then G1 is called a factor of G2.

Recall [4] that the strong product of a pair of graphs G1 = (V1, E 1, ) and G2 = (V2, E 2, ) is the the graph G1⊗G2 = (V1 × V2, E) with the set of edges E = {{(x1, x2), (y1, y2)} ⊆ V1 × V2; there exists a nonempty subset J ⊆ {1, 2} such that {xj , yj} ∈ Ej for every j ∈ J and xj = yj for every j ∈ {1, 2}J}. Note that the strong product differs from the cartesian product of G1 and G2, i.e., from the graph (V1 × V2, F) where F = {{(x1, x2), (y1, y2)}; {xj , yj} ∈ Ej for every j ∈ {1, 2}} (the cartesian product is a factor of the strong product).

By a closure operator u on a set X, we mean a map u: exp X → exp X (where exp X denotes the power set of X) which is

1. grounded (i.e., u∅ = ∅),

2. extensive (i.e., A ⊆ X ⇒ A ⊆ uA), and

3. monotone (i.e., A ⊆ B ⊆ X ⇒ uA ⊆ uB).

The pair (X, u) is then called a closure space. Closure spaces were studied by E. Čech [5] (who called them topological spaces).

A closure operator u on X that is

4. additive (i.e., u(A ∪ B) = uA ∪ uB whenever A, B ⊆ X) and

5. idempotent (i.e., uuA = uA whenever A ⊆ X) is called a Kuratowski closure operator or a topology and the pair (X, u) is called a topological space.

Given a cardinal m > 1, a closure operator u on a set X and the closure space (X, u) are called an Sm- closure operator and an Sm-closure space (briefly, an Sm-space), respectively, if the following condition is satisfied:

A ⊆ X ⇒ uA = ∪ {uB; B ⊆ A, card B < mg.

In [6], S2-closure operators and S2-closure spaces are called quasi-discrete. S2-topologies (S2-topological spaces) are usually called Alexandroff topologies (Alexandroff spaces) - see [7]. Every S2-closure operator is of course additive and every Sα-closure operator is an Sβ-closure operator whenever α < β. If α@0, then every additive Sα-closure operator is an S2-closure operator.

Many concepts defined for topological spaces (see, e.g., [8]) can naturally be extended to closure spaces. Let us mention some of them. Given a closure space (X, u), a subset A ⊆ X is called closed if uA = A, and it is called open if XA is closed. A closure space (X, u) is said to be a subspace of a closure space (Y, v) if uA = vA \ X for each subset A ⊆ X.We will briefly speak about a subspace X of (Y, v). A closure space (X, u) is said to be connected if ∅ and X are the only clopen (i.e., both closed and open) subsets of X. A subset X ⊆ Y is connected in a closure space (Y, v) if the subspace X of (Y, v) is connected. A maximal connected subset of a closure space is called a component of this space. The connectedness graph of a closure operator u on a set X is the graph with the vertex set X whose edges are the connected two-element subsets of X. All the basic properties of connected sets and components in topological spaces are also preserved in closure spaces. In particular, we will employ the fact that the union of a sequence (finite or infinite) of connected subsets is connected if every pair of consecutive members of the sequence has a nonempty intersection.

If u, v are closure operators on a set X, then we put uv if uA ⊆ vA for every subset A ⊆ X (clearly, ≤ is a partial order on the set of all closure operators on X).

In this note, the concept of a Galois connection is understood in the isotone sense. Thus, a Galois connection between partially ordered sets G = (G, ≤) and H = (H, ≤) is a pair (f , g) where f : GH and g : HG are isotone (i.e., order preserving) maps such that f (g(y)) ≤ y for every yH and xg(f (x)) for every x 2 G. For more details concerning Galois connections see [9].

1 Closure operators associated with sets of paths

In the sequel, n will denote a positive integer.

Given a graph G, we denote by Pn(G) the set of all paths of length n in G. For every subset B Pn(G), we put B̂ = {(xi| im); 0 < mn and there exists (yi| in) ∈ B such that xi = yi for every im} (so that the elements of B̂ are the paths of lengths less than or equal to n that are initial parts of the paths belonging to B; we clearly have B B̂).

Let Gj = (Vj , Ej) be a graph and Bj Pn(Gj) for every j = 1, 2. Then, we put ${\mathcal{B}}_{1}\otimes {\mathcal{B}}_{2}=\left\{\left(\left({x}_{i}^{1},{x}_{i}^{2}\right)|\phantom{\rule{thinmathspace}{0ex}}i\le$n); $\left({x}_{i}^{1},{x}_{i}^{2}\right)\in {V}_{1}×{V}_{2}$for all in, there is a nonempty subset J ⊆ {1, 2} such that $\left({x}_{i}^{j}|\phantom{\rule{thinmathspace}{0ex}}i\le n\right)\in {\mathcal{B}}_{j}$for every j ∈ J, and $\left({x}_{i}^{j}|\phantom{\rule{thinmathspace}{0ex}}i\le n\right)$is a constant sequence for every j ∈ {1, 2}J}. We will need the obvious fact that B1⊗B2 Pn(G1⊗G2).

Let G = (V, E) be a graph. Given a subset B Pn(G) (n > 0 an ordinal), we put fn(B)X = X⋃{x ∈ V; there exists (xi| im) B̂ with {xi; i < m} ⊆ X and xm = x} for every X ⊆ V. It may easily be seen that fn(B) is an Sn+1-closure operator on G. Thus, denoting by U(G) the set of all closure operators on G (i.e., on the vertex set V of G), we get a map fn : exp Pn(G) → U(G). The closure operator fn(B) is said to be associated to B. It is evident that every path belonging to B̂ is a connected subset of the closure space (V, fn(B)).

Further, given a closure operator u on G, we put gn(u) = {(xi| in) 2 Pn(G); xj ∈ u{xi; i < j} for every j with 0 < jn}. Thus, we get a map gn : U(G) →exp Pn(G).

Theorem 1.1

Let G = (V, E) be a graph. Then, the pair (fn , gn) constitutes a Galois connection between the partially ordered sets (exp Pn(G), ) and (U(G), ≤).

Proof. It is evident that fn and gn are isotone. Let B Pn(G) and let (yi| in) B be a path. Put fn(B) = u. Then, u is the closure operator on G given by uX = X [ {x ∈ V; there exists (xi| im) B̂ with {xi; i < mg ⊆ X and xm = x} for every X ⊆ V. We have (yi| ij) B̂ for every j, 0 < jn, hence yj ∈ u{yi; i < j}. Consequently, (yi| in) ∈ gn(u). Therefore, B ⊆ gn(u) = gn(fn(B)).

Let u ∈ U(G) and let X ⊆ V be a subset. Put gn(u) = B and let y ∈ fn(gn(u))X be an element. If y ∈ X, then y ∈ uX. Let y ∉ X. Then, there exists (xi| im) B̂ such that {xi; i < m} ⊆ X and xm = y. Since (xi| im) B̂ , there is (yi| i < n) B = gn(u) with xi = yi for every im. Thus, yj ∈ u{xi; i < j} for every j, 0 < j < n. In particular, y = ym ∈ u{xi; i < m} ⊆ uX. Therefore, fn(gn(u)) ≤ u. □

In what follows, we will investigate, for a given graph G, the (isomorphic) partially ordered sets fn(exp Pn(G)) and gn(U(G)). If the graph G is complete, then of course f1(exp P1(G)) is the set of all S2-closure operators on G.

Proposition 1.2

Let G = (V, E) be a graph and B Pn(G). Then, B ∈ gn(U(G)) if and only if the following condition is satisfied:

(*) If (xi| in) Pn(G) has the property that, for every i0, 0 < i0n, there exist (yj| jn) B and j0, 0 < j0n, such that xi0 = yj0 and {yj; j < j0} ⊆ {xi; i < i0}, then (xi| in) B.

Proof. Let B 2 gn(U(G)), let (xi| in) P(G), and let, for any i0, 0 < i0n, there be (yj|jn) B and j0, 0 < j0n, such that xi0 = yj0 and {yj; j < j0} ⊆ {xi; i < i0}. Then, xi0 ∈ fn(B){yj; j < j0} ⊆ fn(B){xi; i < i0} for every i0, 0 < i0n. Therefore, (xij in) ∈ gn(fn(B)) = B. Thus, the condition (*) is satisfied.

Conversely, let the condition (*) be satisfied and let (xi|in) ∈ gn(fn(B)). Then, xi0 ∈ fn(B)fxi; i < i0} for each i0, 0 < i0/ = n. Hence, for every i0, 0 < i0n, there exist (yj| jn) B and j0, 0 < j0n, such that xi0 = yj0 and {yj; j < j0} ⊆ {xi; i < i0}. Therefore, (xi| in) B and we have shown that gn(fn(B)) B. Consequently, B = gn(fn(B)), so that B ∈ gn(U(G)). The proof is complete. □

Example 1.3

Note that every subset B P1(G) satisfies the condition (*). A subset B P2(G) satisfies (*) if and only if each of the following six conditions implies (x, y, z) B:

1. (x, y, t) B, (x, z, u) B,

2. (x, y, t) B, (y, z, u) B,

3. (x, y, t) B, (y, x, z) B.

The following assertion is obvious:

Proposition 1.4

Let G = (V, E) be a graph and u ∈ U(G). Then u ∈ fn(exp Pn(G)) if and only if the following condition is satisfied:

If X ⊆ V and x 2 uXX, then there exist (xi|in) Pn(G) and a positive integer mn, such that {xi; i < m} ⊆ X, xj ∈ u{xi; i < j} for each j, 0 < jn, and xm = x.

Though fn(B) is neither additive nor idempotent in general, we have:

Proposition 1.5

Let G = (V, E) be a graph and B Pn(G) a subset. The union of a system of closed subsets of (V, fn(B)) is a closed subset of (V, fn(B)).

Proof. Put fn(B) = u. Let {Xj; j ∈ J} be a system of closed subsets of (V, u) and let x 2 u ∪ j∈J Xj. Then, there are (xi| in) B and i0, 0 < i0n, such that xi0 = x and xi ∈ ∪ j∈J Xj for all ii0. In particular, we have x0 j∈J Xj so that there is j0 ∈ J such that x0 ∈ Xj0 . If there exists k, 0 < kn, such that xi ∈ Xj0 for all i < k, then xk ∈ u{xi; i < kg ⊆ uXj0 = Xj0 . Consequently, we have {xi; in} ⊆ Xj0 . Thus, x = xi0 ∈ Xj0 ⊆ ∪ j∈J Xj. We have shown that u ∪ j∈J X j ⊆ ∪ j∈J X j, which completes the proof. □

Proposition 1.6

Let G = (V, E) be a graph and B Pn(G) a subset. Then, the closure operator fn(B) is idempotent if and only if (V, fn(B)) is an Alexandroff space.

Proof. Put fn(B) = u. Let u be idempotent and let X ⊆ V be a subset and x 2 u a point. If x 2 X, then x 2 u{x} ⊆ ∪ y∈X u{y}. Suppose that x ∉ X. Then there are (xi| in) B and i0, 0 < i0n, such that xi0 = x and xi ∈ X for all ii0. Clearly, we have x ∈ u{xi; i < i0}. If i0 = 1, then x 2 u{x0}. Let i0 > 1. We have u{xi; i < k} ⊆ uu{xi; i < k − 1} = u{xi; i < k − 1} for every k, 1 < kn. Consequently, u{xi; i < i0} ⊆ u{xi; i < i0 − 1} ⊆ ... ⊆ u{x0}, so that x ∈ u{x0}. Therefore, x ∈ ∪ y∈X u{y} and the proof is complete. □

2 Sets of paths in and the associated closure operators on the 2-adjacency graph

Recall that the 2-adjacency graph (on Z) is the graph Z2 = (Z, A2) where A2 = {{p, q}; p, q ∈ Z, |pq| = 1}. For every l ∈ Z, we put

$Il=(ln+i|i≤n) if l is odd,((l+1)n−i|i≤n) if l is even.$

Throughout this section, B Pn(Z2) will denote the set B = {Il; l ∈ Z}. Thus, all paths Il belonging to B are just the arithmetic sequences (xi|in) of integers with the difference equal to 1 or −1 and with x0 = ln if l is odd and x0 = (l + 1)n if l is even. Note that each element z 2 Z belongs to at least one and at most two paths in B. It belongs to two (different) paths from B if and only if there is l 2 Z with z = ln (in which case z is the first member of each of the paths Il and Il−1 if l is odd, and z is the last member of each of the two paths if l is even). The graph Z2 with the set B of paths is demonstrated in Figure 1 where only the vertices kn, k 2 Z, are marked (by bold dots) so that, between any two neighboring vertices marked, there are n − 1 more vertices that are not marked out. The paths in B are represented by the line segments oriented from the first to the last members of the paths (and every directed line segment represents n edges of the graph).

Clearly, the closure operator fn(B) is additive if and only if n = 1. The closure operator f1(B) coincides with the Khalimsky topology on Z generated by the subbase {{2k − 1, 2k, 2k + 1 }; k ∈ Z} - cf. [1].

Figure 1

A section of the graph Z2 with the set B of paths demonstrated.

Proposition 2.1

In the closure space (Z, fn(B)), the points ln, l ∈ Z odd, are open while all the other points are closed.

Proof. Let l0 Z be an arbitrary odd number and put z = l0n. Let x 2 fn(B)(Z − {z}) be a point and suppose that x ∉ Z − {z}, i.e., that x = z. Then, there are (xi| in) B and i0, 0 < i0n, such that x = xi0 and {xi; i < i0} ∈ Z − {z}. Let m ∈ Z be the odd number with xi = mn + i for all in or xi = mni for all in. Then, xi0 = mn + i0 or xi0 = mni0. Since 0 < i0n, we have 0 < i0 < 2n and, consequently, mn < xi0 < (m + 2)n or (m − 2)n < xi0 < mn. Hence, xi0ln for any odd number l ∈ Z. Thus, xi0z, so that x ∈ Z − {z}. Therefore, the set Z − {z} is closed, i.e., {z} is open in (Z, fn(B)).

Let z ∈ Z be an arbitrary point with zln for any odd number l ∈ Z. Let x ∈ fn(B){z} and suppose that xy. Then, there are (xi| in) B and i0, 0 < i0n, such that x = xi0 and {xi; i < i0} = {z}. Thus, x0 = z, which contradicts the existence of an odd number l0 Z with x0 = l0n. Hence, x = y and, therefore, {z} is closed in (Z, fn(B)). □

Theorem 2.2

(Z, fn(B)) is a connected closure space.

Proof. Since every path belonging to B is connected in (Z, fn(B)), the set A = I0 ∪ I1 ∪ I2 ... of non-negative integers and the set B = I−1 ∪ I−2 ∪ I−3 ... of non-positive integers are connected (note that Ik ∩ Ik+1 = {kn}).

Thus, Z = A ∪ B is connected because A ∩ B = {0}. □

Given a totally ordered set (X, ≤) and a point x ∈ X, we put L(x) = {y ∈ Z; y < x} and U(x) = {y ∈ Z; y > x}. The set of integers Z is considered to be totally ordered by the natural order. Clearly, for every z ∈ Z, both L(z) and U(z) are closed in the subspace Z − {z} of (Z, fn(B)).

Theorem 2.3

Let z ∈ Z be a point. Then, there are points z1, z2 Z such that L(z1) and U(z2) are components of the subspace Z − {z} of (Z, fn(B)) and all the other components of Z − {z} are singletons. If z = ln + i where l, i ∈ Z, l even and |i| ≤ 1, then z1 = z2 = z (so that Z − {z} has no singleton components).

Proof. There are l0, i0 Z, l odd and |i0|n, such that z = l0n + i0. Let |i0| = n. Then, z = (l0 + 1)n or z = (l0 − 1)n so that there is an even number m ∈ Z with z = mn. We clearly have L(z) = ⋃ {Il; l ≤ (m−2)n}⋃{i; (m−1)ni < z}. L(z) is connected in (Z, fn(B)) because (Il| l ≤ (m−2)n) is a sequence of paths belonging to B with every pair of consecutive paths having a point in common, {i; (m − 1)ni < z} 2 B̂ and (m − 1)n ∈ I(m−2)n ∩ {i; (m − 1)ni < z}. Similarly, U(z) = ⋃ {Il; l ≥ (m + 1)n} ⋃ {i; z < i ≤ (m + 1)n} is connected in (Z, fn(B)). Since L(z) and U(z) are closed, disjoint, and satisfying L(z) ⋃ U(z) = Z −{z}, they are the components of Z − {z}.

Let |i0| < n and suppose that i0 ≥ 0. Then, L(l0n) is connected in (Z, fn(B)) because it is the union of a sequence of paths belonging to B, namely the sequence (Il| ll0) in which every pair of consecutive paths has a point in common. Further, since (l0n + i| i < l0n + i0) B̂ and L(l0n) ⋂ {l0n + i; i < l0n + i0}, the set L(z) = L(l0n) ∪ {l0n + i; i < l0n + i0} is connected in (Z, fn(B)). It is also evident that L(z) is closed in the subspace Z − {z}. Further, U((l0 + 1)n − 1) is connected because it is the union of a sequence of paths belonging to B, namely, the sequence (Il| ll0 + 1) in which every pair of consecutive paths has a point in common. It is also evident that U(z) is closed in the subspace Z −{z}. Clearly, we have Z −{z} = L(z)∪{i; z < i < 2l0n} ∪ U((l0 + 1)n − 1) where the sets L(z), {i; z < i < 2l0n}, and U((l0 + 1)n − 1) are pairwise disjoint. The singleton subsets of {i; z < i < (l0 + 1)n} are closed in Z − {z}.We have shown that L(z), U((l0 + 1)n − 1),and the singletons {i}, z < i < (l0 + 1)n, are the components of Z − {z}. We may show in an analogous way that U(z), L((l0 + 1)n − 1), and the singletons {i}, (l0 − 1)n < i < z, are the components of Z − {z} if i0 ≤ 0.

To prove the second part of the Theorem, suppose that z = ln + i where l, i ∈ Z, l even and |i| ≤ 1. It was shown in the first part of the proof that L(z) and U(z) are the (only) components of Z − {z} if i = 0 (because then z = l0n + i0 where l0 = l1 is odd and i0 = n). Suppose that i = 1. Then, z = l0n + i0 where l0 = l + 1 and i0 = 1 − n. We have (l0 − 1)n + 1 = ln + 1 = z, so that L((l0 − 1)n + 1) = L(z). Since {i; (l0 − 1)n < i < z} = ∅, L(z) and U(z) are the (only) components of Z − {z} according to the previous part of the proof. Using similar arguments, we may show that L(z) and U(z) are the (only) components of Z − {z} if i = −1. The proof is complete. □

Remark 2.4

Recall [1] that a connected topological space (X, u) is called a connected ordered topological space (COTS for short) if, for any three-point subset Y ⊆ X, there is a point x 2 Y such that Y meets two components of the subspace X{x}. It was shown in [1] that a connected topological space (X, u) is a COTS if and only if there is a total order on X such that, for every x ∈ X, the sets L(x) and U(x) are components of the subspace X{x}. Thus, Theorem 3 results in the well-known fact that the Khalimsky space (Z, f1(B)) is a COTS ([1]). Hence, the closure operators fn(B) may be regarded as generalizations of the connected ordered topologies on Z.

3 Closure operators associated with sets of paths in the digital plane

In the sequel, we will discuss the graph Z2 ⊗ Z2 with the set B ⊗ B of paths of length n. The graph is demonstrated in Figure 2 where, as in Figure 1, only the vertices (kn, ln), k, l ∈ Z, are marked (by bold dots) and only the paths between these vertices are marked (by line segments oriented from the first to the last members of the paths). Thus, between any pair of neighboring parallel horizontal or vertical line segments (having the same orientation), there are n − 1 more parallel line segments with the same orientation that are not displayed in order to make the Figure transparent. And, of course, every oriented line segment represents n edges of the graph.

Figure 2

A section of the graph Z2⊗Z2 with the set B⊗B of paths demonstrated.

Observe that the closure space (Z2, fn(B⊗B)) is connected. Indeed, the set Z ×{k} is connected for every k 2 Z by applying arguments similar to those used in the proof of Theorem 2 Therefore, the set Z2 = A ∪ B where A = (Z × {0}) (Z × {1}) (Z × {2}) ... and B = (Z × {0}) (Z × {−1}) (Z × {−2}) ... is connected, again, by applying arguments similar to those used in the proof of Theorem 2.

Note that Z 2Z 2 is nothing but the well-known 8-adjacency graph on Z2. Let H n denote the factor of the graph Z2⊗Z2 with exactly those edges {(x1, y1), (x2, y2)} of Z2⊗Z2 that satisfy one of the following four conditions for some k ∈ Z:

x1y1 = x2y2 = 2kn,

x1 + y1 = x2 + y2 = 2kn,

x1 = x2 = 2kn,

y1 = y2 = 2kn.

A section of the graph H n is shown in Figure 3 where only the vertices (2kn, 2ln), k, l ∈ Z, are marked (by bold dots) and thus, on every edge drawn between two such vertices, there are 2n − 1 more (non-displayed) vertices so that the edges represent 2n edges in the graph Hn. Note that every circle C in Hn is a connected subset of (Z2, fn(B⊗B)). Indeed, C consists (is the union) of a finite sequence of paths in Z2⊗Z2 belonging to B B such that every pair of consecutive paths in the sequence has a point in common.

Figure 3

A section of the graph Hn.

The closure operator f1(B⊗B) (which is a topology) is called the Khalimsky topology on Z2 and the topological space (Z2, f1(B⊗B)) is called the Khalimsky plane (cf. [1]). The connectedness graph of the Khalimsky topology is demonstrated in Figure 4.

Figure 4

A section of the connectedness graph of the Khalimsky topology on Z2.

It is a basic problem of digital geometry (cf. [10]) to find a structure on the digital plane Z2 convenient for the study of digital images. The convenience means that such a structure satisfies parallels of some basic geometric and topological properties of the Euclidean topology on the real plane R2. Of these parallels, the validity of an analogue of the Jordan curve theorem plays a crucial role because digital Jordan curves represent the borders of objects in digital images - see [11, 12]. It is well known that the Khalimsky topology provides such a structure on Z2 (cf. [1]). But it was shown in [13, 14] that there are some other topologies and closure operators on Z2 with this property.

According to [1], a circle C in the connectedness graph of the Khalimsky topology f1(B⊗B) is said to be a Jordan curve in the Khalimsky plane if the following two conditions are satisfied:

1. with each of its points, C contains precisely two points adjacent to it,

2. C separates the Khalimsky plane into exactly two components (i.e., the subspace Z2C of (Z2, f1(B⊗B)) consists of exactly two components)..

As an immediate consequence of Theorem 5.6 proved in [1], we get the following digital Jordan curve theorem for the Khalimsky space (Z2, f1(B⊗B)): A circle C in the graph H1 is a Jordan curve in the Khalimsky plane if and only if, at none of its points, it turns at an acute angle of $\frac{\pi }{4}.$

For every n > 1, we define (digital) Jordan curves in (Z2, fn(B⊗B)) to be the circles C in Hn that separate (Z2, fn(B⊗B)) into exactly two components.

Now the problem arises to determine, for every n > 1, those circles in H n that are Jordan curves in (Z2, fn(B⊗B)). The following solution of the problem results from Theorem 3.19 proved in [15] (in quite a laborious way based on using quotient closure spaces and the Jordan curve theorem for the Khalimsky plane, namely, Theorem 5.6 in [1]):

Theorem 3.1

Every circle in Hn that does not turn at any point ((2k+1)n, (2l+1)n), k, l ∈ Z, is a Jordan curve in (Z2, fn(B⊗B)) whenever n > 1.

Thus, the closure operators fn(B⊗B), n > 1, may be used as background structures on Z2 for the study of digital images. The advantage of using these closure operators rather than the Khalimsky topology f1(B⊗B) is that the Jordan curves with respect to them, i.e., circles in H n, may turn at an acute angle of $\frac{\pi }{4}$at some points - see the following example:

Example 3.2

Consider the set of points of Z2 demonstrated in Figure 5, which represents the (border of) letter M (the points may be regarded as centers of pixels in a computer screen). This set is not a Jordan curve in the Khalimsky plane (Z2, f1(B⊗B)). For it to be a Jordan curve in the Khalimsky plane, the eight points that are ringed have to be deleted. But this would lead to a certain deformation (loss of sharpness) of the letter - the eight pixels will belong to the white background of the black image of M. On the other hand, the set is a circle in the graph H 2 satisfying the assumption of Theorem 4 and, therefore, it is a Jordan curve in (Z2, f2(B⊗B)).

Figure 5

A digital image of M.

Acknowledgement

This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic from the National Programme of Sustainability (NPU II) project "IT4Innovations Excellence in Science - LQ1602".

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Accepted: 2018-12-06

Published Online: 2018-12-31

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1573–1581, ISSN (Online) 2391-5455,

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