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

Josef Šlapal

doi:10.1515/math-2018-0128

Article Open Access

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

Josef Šlapal

Published/Copyright: December 31, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

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.

Keywords: Simple graph; Closure operator; Galois connection; digital space; Khalimsky topology; Jordan curve theorem

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 V ≠ Ø as the vertex set and E ⊆ {{x, y}; x, y ∈ V, x ≠ y} 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 (x_i| i ≤ n), i.e., (x₀, x₁, ..., x_n), of pairwise different vertices of V such that x_i is adjacent to x_i₊₁ whenever i < n. The non-negative integer n is called the length of the path (x_i| i ≤ n). A sequence (x_i| i ≤ n) of vertices of G is called a circle if n > 2, x₀ = x_n, and (x_i| 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⊆ E₂. 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, E₁, ) and G₂ = (V₂, E₂, ) is the the graph G_1⊗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 {x_j , y_j} ∈ E_j for every j ∈ J and x_j = y_j 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)}; {x_j , y_j} ∈ E_j 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

grounded (i.e., u∅ = ∅),
extensive (i.e., A ⊆ X ⇒ A ⊆ uA), and
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
additive (i.e., u(A ∪ B) = uA ∪ uB whenever A, B ⊆ X) and
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 S_m- closure operator and an S_m-closure space (briefly, an S_m-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 S_α-closure operator is an S_β-closure operator whenever α < β. If α ≤ @₀, then every additive S_α-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 X − A 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 u ≤ v 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 : G → H and g : H → G are isotone (i.e., order preserving) maps such that f (g(y)) ≤ y for every y ∈ H and x ≤ g(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 P_n(G) the set of all paths of length n in G. For every subset B ⊆ P_n(G), we put B̂ = {(x_i| i ≤ m); 0 < m ≤ n and there exists (y_i| i ≤ n) ∈ B such that x_i = y_i for every i ≤ m} (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 G_j = (V_j , E_j) be a graph and B_j ⊆ P_n(G_j) for every j = 1, 2. Then, we put B1⊗B2={((xi1,xi2)|i≤n); (xi1,xi2)∈V1×V2for all i ≤ n, there is a nonempty subset J ⊆ {1, 2} such that (xij|i≤n)∈Bjfor every j ∈ J, and (xij|i≤n)is a constant sequence for every j ∈ {1, 2} − J}. We will need the obvious fact that B_1⊗B₂⊆ P_n(G_1⊗G₂).

Let G = (V, E) be a graph. Given a subset B ⊆ P_n(G) (n > 0 an ordinal), we put f_n(B)X = X⋃{x ∈ V; there exists (x_i| i ≤ m) ∈ B̂ with {x_i; i < m} ⊆ X and x_m = x} for every X ⊆ V. It may easily be seen that f_n(B) is an S_n₊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 f_n : exp P_n(G) → U(G). The closure operator f_n(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, f_n(B)).

Further, given a closure operator u on G, we put g_n(u) = {(x_i| i ≤ n) 2 P_n(G); x_j ∈ u{x_i; i < j} for every j with 0 < j ≤ n}. Thus, we get a map g_n : U(G) →exp P_n(G).

Theorem 1.1

Let G = (V, E) be a graph. Then, the pair (f_n , g_n) constitutes a Galois connection between the partially ordered sets (exp P_n(G), ⊆) and (U(G), ≤).

Proof. It is evident that f_n and g_n are isotone. Let B ⊆ P_n(G) and let (y_i| i ≤ n) ∈ B be a path. Put f_n(B) = u. Then, u is the closure operator on G given by uX = X [ {x ∈ V; there exists (x_i| i ≤ m) ∈ B̂ with {x_i; i < mg ⊆ X and x_m = x} for every X ⊆ V. We have (y_i| i ≤ j) ∈ B̂ for every j, 0 < j ≤ n, hence y_j ∈ u{y_i; i < j}. Consequently, (y_i| i ≤ n) ∈ g_n(u). Therefore, B ⊆ g_n(u) = g_n(f_n(B)).

Let u ∈ U(G) and let X ⊆ V be a subset. Put g_n(u) = B and let y ∈ f_n(g_n(u))X be an element. If y ∈ X, then y ∈ uX. Let y ∉X. Then, there exists (x_i| i ≤ m) ∈ B̂ such that {x_i; i < m} ⊆ X and x_m = y. Since (x_i| i ≤ m) ∈ B̂ , there is (y_i| i < n) ∈ B = g_n(u) with x_i = y_i for every i ≤ m. Thus, y_j ∈ u{x_i; i < j} for every j, 0 < j < n. In particular, y = y_m ∈ u{x_i; i < m} ⊆ uX. Therefore, f_n(g_n(u)) ≤ u. □

In what follows, we will investigate, for a given graph G, the (isomorphic) partially ordered sets f_n(exp P_n(G)) and g_n(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 ⊆ P_n(G). Then, B ∈ g_n(U(G)) if and only if the following condition is satisfied:

(*) If (x_i| i ≤ n) ∈ P_n(G) has the property that, for every i₀, 0 < i₀ ≤ n, there exist (y_j| j ≤ n) ∈ B and j₀, 0 < j₀ ≤ n, such that x_i₀ = y_j₀and {y_j; j < j₀} ⊆ {x_i; i < i₀}, then (x_i| i ≤ n) ∈ B.

Proof. Let B 2 g_n(U(G)), let (x_i| i ≤ n) ∈ P₍G), and let, for any i₀, 0 < i₀ ≤ n, there be (y_j|j ≤ n) ∈ B and j₀, 0 < j₀ ≤ n, such that x_i₀ = y_j₀ and {y_j; j < j₀} ⊆ {x_i; i < i₀}. Then, x_i₀∈ f_n(B){y_j; j < j₀} ⊆ f_n(B){x_i; i < i₀} for every i₀, 0 < i₀ ≤ n. Therefore, (x_ij i ≤ n) ∈ g_n(f_n(B)) = B. Thus, the condition (*) is satisfied.

Conversely, let the condition (*) be satisfied and let (x_i|i ≤ n) ∈ g_n(f_n(B)). Then, x_i₀∈ f_n(B)fx_i; i < i₀} for each i₀, 0 < i_0/ = n. Hence, for every i₀, 0 < i₀ ≤ n, there exist (y_j| j ≤ n) ∈ B and j₀, 0 < j₀ ≤ n, such that x_i₀ = y_j₀ and {y_j; j < j₀} ⊆ {x_i; i < i₀}. Therefore, (x_i| i ≤ n) ∈ B and we have shown that g_n(f_n(B)) ⊆ B. Consequently, B = g_n(f_n(B)), so that B ∈ g_n(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:

(x, y, t) ∈ B, (x, z, u) ∈ B,
(x, y, t) ∈ B, (y, z, u) ∈ B,
(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 ∈ f_n(exp P_n(G)) if and only if the following condition is satisfied:

If X ⊆ V and x 2 uX − X, then there exist (x_i|i ≤ n) ∈ P_n(G) and a positive integer m ≤ n, such that {x_i; i < m} ⊆ X, x_j ∈ u{x_i; i < j} for each j, 0 < j ≤ n, and x_m = x.

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

Proposition 1.5

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

Proof. Put f_n(B) = u. Let {X_j; j ∈ J} be a system of closed subsets of (V, u) and let x 2 u ∪ _j∈J X_j. Then, there are (x_i| i ≤ n) ∈ B and i₀, 0 < i₀ ≤ n, such that x_i₀ = x and x_i ∈ ∪ _j∈J X_j for all i ≤ i₀. In particular, we have x₀∈ _j∈J X_j so that there is j₀∈ J such that x₀∈ X_j₀ . If there exists k, 0 < k ≤ n, such that x_i ∈ X_j₀ for all i < k, then x_k ∈ u{x_i; i < kg ⊆ uX_j₀ = X_j₀ . Consequently, we have {x_i; i ≤ n} ⊆ X_j₀ . Thus, x = x_i₀∈ X_j₀⊆ ∪ _j∈J X_j. 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 ⊆ P_n(G) a subset. Then, the closure operator f_n(B) is idempotent if and only if (V, f_n(B)) is an Alexandroff space.

Proof. Put f_n(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 (x_i| i ≤ n) ∈ B and i₀, 0 < i₀ ≤ n, such that x_i₀ = x and x_i ∈ X for all i ≤ i₀. Clearly, we have x ∈ u{x_i; i < i₀}. If i₀ = 1, then x 2 u{x₀}. Let i₀ > 1. We have u{x_i; i < k} ⊆ uu{x_i; i < k − 1} = u{x_i; i < k − 1} for every k, 1 < k ≤ n. Consequently, u{x_i; i < i₀} ⊆ u{x_i; 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, |p − q| = 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 ⊆ P_n(Z₂) will denote the set B = {I_l; l ∈ Z}. Thus, all paths I_l belonging to B are just the arithmetic sequences (x_i|i ≤ n) 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 I_l and I_l₋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 f_n(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].

$Figure 1 A section of the graph Z2 with the set B of paths demonstrated.$

Figure 1

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

Proposition 2.1

In the closure space (Z, f_n(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 f_n(B)(Z − {z}) be a point and suppose that x ∉ Z − {z}, i.e., that x = z. Then, there are (x_i| i ≤ n) ∈ B and i₀, 0 < i₀ ≤ n, such that x = x_i₀ and {x_i; i < i₀} ∈ Z − {z}. Let m ∈ Z be the odd number with x_i = mn + i for all i ≤ n or x_i = mn − i for all i ≤ n. Then, x_i₀ = mn + i₀ or x_i₀ = mn − i₀. Since 0 < i₀ ≤ n, we have 0 < i₀ < 2n and, consequently, mn < x_i₀ < (m + 2)n or (m − 2)n < x_i₀ < mn. Hence, x_i₀ ≠ ln for any odd number l ∈ Z. Thus, x_i₀ ≠ z, so that x ∈ Z − {z}. Therefore, the set Z − {z} is closed, i.e., {z} is open in (Z, f_n(B)).

Let z ∈ Z be an arbitrary point with z ≠ln for any odd number l ∈ Z. Let x ∈ f_n(B){z} and suppose that x ≠ y. Then, there are (x_i| i ≤ n) ∈ B and i₀, 0 < i₀ ≤ n, such that x = x_i₀ and {x_i; 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, f_n(B)). □

Theorem 2.2

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

Proof. Since every path belonging to B is connected in (Z, f_n(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 I_k ∩ I_k₊₁ = {k_n}).

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, f_n(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, f_n(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) = ⋃ {I_l; l ≤ (m−2)n}⋃{i; (m−1)n ≤ i < z}. L(z) is connected in (Z, f_n(B)) because (I_l| 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)n ≤ i < z} 2 B̂ and (m − 1)n ∈ I_(m−2)n∩ {i; (m − 1)n ≤ i < z}. Similarly, U(z) = ⋃ {I_l; l ≥ (m + 1)n} ⋃ {i; z < i ≤ (m + 1)n} is connected in (Z, f_n(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, f_n(B)) because it is the union of a sequence of paths belonging to B, namely the sequence (I_l| l ≤ l₀) 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, f_n(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 (I_l| l ≥ l₀ + 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 f_n(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 Z_{2 ⊗} 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.

$Figure 2 A section of the graph Z2⊗Z2 with the set B⊗B of paths demonstrated.$

Figure 2

A section of the graph Z_2⊗Z₂ with the set B⊗B of paths demonstrated.

Observe that the closure space (Z², f_n(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 Z_2⊗Z₂ with exactly those edges {(x₁, y₁), (x₂, y₂)} of Z_2⊗Z₂ that satisfy one of the following four conditions for some k ∈ Z:

x₁ − y₁ = x₂ − y₂ = 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 H_n. Note that every circle C in H_n is a connected subset of (Z², f_n(B⊗B)). Indeed, C consists (is the union) of a finite sequence of paths in Z_2⊗Z₂ belonging to B B such that every pair of consecutive paths in the sequence has a point in common.

$Figure 3 A section of the graph Hn.$

Figure 3

A section of the graph H_n.

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.

$Figure 4 A section of the connectedness graph of the Khalimsky topology on Z2.$

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:

with each of its points, C contains precisely two points adjacent to it,
C separates the Khalimsky plane into exactly two components (i.e., the subspace Z² −C 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 π4.

For every n > 1, we define (digital) Jordan curves in (Z², f_n(B⊗B)) to be the circles C in H_n that separate (Z², f_n(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, f_n(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 H_n that does not turn at any point ((2k+1)n, (2l+1)n), k, l ∈ Z, is a Jordan curve in (Z², f_n(B⊗B)) whenever n > 1.

Thus, the closure operators f_n(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 π4at 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 H₂ satisfying the assumption of Theorem 4 and, therefore, it is a Jordan curve in (Z², f₂(B⊗B)).

$Figure 5 A digital image of M.$

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".

References

[1] Khalimsky E.D., Kopperman R., Meyer P.R., Computer graphics and connected topologies on finite ordered sets, Topology Appl., 1990, 36, 1–1710.1016/0166-8641(90)90031-VSearch in Google Scholar

[2] Šlapal J., Graphs with a path partition for structuring digital spaces, Inf. Sciences, 2013, 233, 305–31210.1016/j.ins.2013.01.011Search in Google Scholar

[3] Harrary F., Graph Theory, Addison-Wesley Publ. Comp., Reading, Massachussets, Menlo Park, California, London, Don Mills, Ontario, 1969Search in Google Scholar

[4] Sabidussi G., Graph multiplication, Math. Z., 1960, 72, 446-45710.1007/BF01162967Search in Google Scholar

[5] Cech E., Topological spaces, In: Topological Papers of Eduard Cech, Academia, Prague, 1968, ch. 28, 436–472Search in Google Scholar

[6] Cech E., Topological Spaces (revised by Z. Frolík and M. Katětov), Academia, Prague, 1966Search in Google Scholar

[7] Kong T.Y, Kopperman R., Meyer P.R., A topological approach to digital topology, Amer. Math. Monthly, 1991, 98, 902–91710.1080/00029890.1991.12000810Search in Google Scholar

[8] Engelking R., General Topology, Panstwowe Wydawnictwo Naukowe, Warszawa, 1977Search in Google Scholar

[9] Davey B.A., Priestley H.A., Introduction to Lattices and Order, Cambridge University Press, Cambridge, 200210.1017/CBO9780511809088Search in Google Scholar

[10] Klette R., Rosenfeld A., Digital Geometry - Geometric Methods for Digital Picture Analysis, Elsevier, Singapore, 2006Search in Google Scholar

[11] Brimkov V.E., Klette R., Border and surface tracing - theoretical foundations, IEEE Trans. Patt. Anal. Machine Intell., 2008, 30, 577–59010.1109/TPAMI.2007.70725Search in Google Scholar PubMed

[12] Khalimsky E.D., Kopperman R., Meyer P.R., Boundaries in digital plane, J. Appl. Math. Stochast. Anal., 1990, 3, 27–5510.1155/S1048953390000041Search in Google Scholar

[13] Šlapal J., Jordan curve theorems with respect to certain pretopologies on Z² Lect. Notes in Comput. Sci., 2009, 5810, 252–26210.1007/978-3-642-04397-0_22Search in Google Scholar

[14] Šlapal J., Convenient closure operators on Z² Lect. Notes in Comput. Sci., 2009, 5852, 425–43610.1007/978-3-642-10210-3_33Search in Google Scholar

[15] Šlapal J., A digital analogue of the Jordan curve theorem, Discr. Appl. Math., 2004, 139, 231–25110.1016/j.dam.2002.11.003Search in Google Scholar

Received: 2018-07-31

Accepted: 2018-12-06

Published Online: 2018-12-31

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0128

Keywords for this article

Simple graph; Closure operator; Galois connection; digital space; Khalimsky topology; Jordan curve theorem

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