Path-induced closure operators on graphs for defining digital Jordan surfaces

Josef Šlapal

doi:10.1515/math-2019-0121

Article Open Access

Path-induced closure operators on graphs for defining digital Jordan surfaces

Josef Šlapal

Published/Copyright: November 19, 2019

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

Given a simple graph with the vertex set X, we discuss a closure operator on X induced by a set of paths with identical lengths in the graph. We introduce a certain set of paths of the same length in the 2-adjacency graph on the digital line ℤ and consider the closure operators on ℤ^m (m a positive integer) that are induced by a special product of m copies of the introduced set of paths. We focus on the case m = 3 and show that the closure operator considered provides the digital space ℤ³ with a connectedness that may be used for defining digital surfaces satisfying a Jordan surface theorem.

Keywords: simple graph; path; closure operator; connectedness; digital space; digital surface; Khalimsky topology; Jordan surface theorem

MSC 2010: 54A05; 54D05; 52C99; 68R10

1 Introduction

In digital picture analysis, detection of object borders plays an important role in solving numerous problems such as pattern recognition etc. - cf. [1]. In two-dimensional digital pictures, it is required that object borders be digital Jordan curves, i.e., subsets of the digital plane ℤ² satisfying a digital analog of the Jordan curve theorem. (Recall that the classical Jordan curve theorem states that a simple closed curve separates the Euclidean plane into precisely two connected components). It is, therefore, necessary to equip the digital plane with a connectedness structure making it possible to define digital Jordan curves. In the classical approach to this problem, a pair of adjacency relations (4- and 8-adjacency) on ℤ² is employed (see [2, 3]). In [4], a new, topological approach was proposed using a single connectedness structure, the so-called Khalimsky topology to provide ℤ² with a connectedness structure. The topological approach was then developed by many authors, see, e.g., [5, 6, 7, 8].

In three-dimensional pictures, object borders are to be digital surfaces, i.e., subsets of the digital space ℤ³ satisfying a digital analog of the Jordan surface theorem (which is also known as the Jordan-Brouwer theorem). A classical approach to this three-dimensional problem is based, like in the two-dimensional case, on using a pair of adjacency relations (6- and 26-adjacency) on ℤ³- see [9, 10, 11, 12]. The topological approach employing the Khalimsky topology on ℤ³ was applied, e.g., in [13, 14].

The present paper is a contribution to the topological approach to the problem of recognizing digital surfaces in ℤ². Instead of the Khalimsky topology, we employ closure operators that are induced by a set of paths of the same length in a simple graph. We introduce, for every positive integer n, a certain set of paths of identical lengths n in the 2-adjacency graph on the digital line ℤ. For every positive integer m, we obtain a closure operator on ℤ^m induced by a special product of m copies of the introduced set of paths in ℤ. For n = 2, we get the well-known Khalimsky topology on ℤ^m. We focus on the case of n = m = 3 and, for the obtained closure operator on ℤ³, we prove a digital Jordan surface theorem.

Sets of paths of identical lengths in a graph were proposed for the study of connectedness in digital spaces in [15]. Closure operators induced by such sets of paths were used in [16] for proving a digital Jordan curve theorem. In [17], correspondences between sets of paths and closure operators in a simple graph were studied. In the present paper, we build on the concepts and results in [17]. To make the paper self-contained, we repeat some of them.

By a graph G = (V, E), we understand an (undirected simple) graph (without loops) where V ≠ ∅ is the vertex set and E ⊆ {{x, y}; x, y ∈ V, x ≠ y} is the set of edges of G. We will say that G is a graph on V. Two vertices x, y ∈ V are said to be adjacent (to each other) if {x, y} ∈ E. Recall that a walk 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. If, moreover, the members of (x_i | i ≤ n) are pairwise different, then (x_i | i ≤ n) is called a path. The non-negative integer n is called the length of the walk (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. For the graph-theoretic background, we refer to [18].

Given graphs G_j = (V_j, E_j,), j = 1, 2, …, m (m > 0 an integer), we define their strong product to be the graph ∏j=1m G_j = ( ∏j=1m V_j, E) with the set of edges E = {{(x₁, x₂, …, x_m), (y₁, y₂, …, y_m)}; there exists a nonempty subset J ⊆ {1, 2, …, m} such that {x_j, y_j} ∈ E_j for every j ∈ J and x_j = y_j for every j ∈ {1, 2, …, m} – J}. Note that the strong product differs from the cartesian product of G_j, j = 1, 2, …, m, i.e., from the graph ( ∏j=1m V_j, F) where F = {{(x₁, x₂, …, x_m), (y₁, y₂, …, y_m)}; {x_j, y_j} ∈ E_j for every j ∈ {1, 2, …, m}}. More precisely, we have F ⊊ E whenever m > 1. The strong product of a pair of graphs coincides with that introduced in [19].

If G_j = G for every j = 1, 2, …, m, we write G^m instead of ∏j=1m G_j.

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.

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 < m}.

S₂-topologies (S₂-topological spaces) are usually called Alexandroff topologies (Alexandroff spaces) - see [7]. Similarly to [17], we will use some basic topological concepts such as closed subsets, subspaces, connected subsets, (connected) components etc. (see, e.g., [20]) naturally extended from topological spaces to closure ones. The behavior of extended concepts is then analogous to that of the original ones. In particular, we will employ the fact that the union of a (finite or infinite) sequence of connected subsets of a closure space is connected in the space if every pair of consecutive members of the sequence has a nonempty intersection.

We will say that a subset Y of a closure space (X, u) separates the space into exactly two components if the subspace X – Y of (X, u) has exactly two components.

2 Closure operators induced by 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 set of paths (path set for short) 𝓑 ⊆ 𝓟_n(G), we put

𝓑̂ = {(x_i | i ≤ m) ∈ 𝓟_m(G); 0 < m ≤ n and there exists (y_i | i ≤ n) ∈ 𝓑 such that x_i = y_i for every i ≤ m} and

𝓑^* = {(x_i | i ≤ m) ∈ 𝓟_m(G); 0 < m ≤ n and (x_i | i ≤ m) ∈ 𝓑̂ or (x_m–i | i ≤ m) ∈ 𝓑̂}.

Thus, we have 𝓑 ⊆ 𝓑̂ ⊆ 𝓑^*.

Let G_j be a graph and 𝓑_j ⊆ 𝓟_n(G_j) for every j = 1, 2, …, m (m > 0 an integer). Then, we put ∏j=1m 𝓑_j = {((xi1,xi2,...,xim)|i≤n); there is a nonempty subset J ⊆ {1, 2, …, m} such that ( xij | i ≤ n) ∈ 𝓑_j for every j ∈ J and ( xij | i < j) is a constant sequence for every j ∈ {1, 2, …, m} – J}. It is evident that ∏j=1m 𝓑_j ⊆ 𝓟_n( ∏j=1m G_j). ∏j=1m 𝓑_j will be called the strong product of 𝓑_j, j = 1, 2, …, m (it will always be clear whether a strong product discussed relates to graphs or path sets). If G_j = G and 𝓑_j = 𝓑 for every j = 1, 2, …, m, we write 𝓑^m instead of ∏j=1m 𝓑_j.

Let G be a graph with the vertex set V and 𝓑 ⊆ 𝓟_n(G). For every X ⊆ V, 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}.

It may easily be seen that f_n(𝓑) is an S_n+1-closure operator on V - it will be said to be associated with 𝓑. It is evident that every path belonging to 𝓑^* is a connected subset of the closure space (V, f_n(𝓑)). For the properties of the closure operators f_n(𝓑) see [17].

Recall [16] that, given a graph G = (V, E) and 𝓑 ⊆ 𝓟_n(G), a sequence C = (x_i | i ≤ p), p > 0 an integer, of vertices of V is called a 𝓑-walk in G if there is an increasing sequence (i_k | k ≤ q), q > 0 an integer, of non-negative integers with i₀ = 0 and i_q = p such that i_k – i_k−1 ≤ n and (x_i | i_k−1 ≤ i ≤ i_k) ∈ 𝓑^* for every k with 0 < k ≤ q. If the terms of C are pairwise different, then C is called a 𝓑-path in G.

A 𝓑-walk C is said to be a 𝓑-circle if, for every pair i₀, i₁ of different integers with 0 ≤ i₀, i₁ ≤ p, x_i₀ = x_i₁ is equivalent to {i₀, i₁} = {0, p}.

Clearly, every 𝓑-walk (𝓑-path, 𝓑-circle) in a graph G = (V, E) is a walk (path, circle) in G and both concepts coincide if 𝓑 = {(x, y); {x, y} ∈ E} ⊆ 𝓟₁(G).

We will need the following statement proved in [16]:

Proposition 2.1

Let G be a graph with the vertex set V and 𝓑 ⊆ 𝓟_n(G). A subset A ⊆ V is connected in the closure space (V, f_n(𝓑)) if and only if any two different vertices of G belonging to A can be joined by a 𝓑-walk in G contained in A.

3 The closure operator on ℤ³ induced by a set of paths of length 2

Recall that the 2-adjacency graph (on ℤ) is the graph H = (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.

In the sequel, (for a given integer n > 1) 𝓑 will denote the set 𝓑 ⊆ 𝓟_n(H) given by 𝓑 = {I_l; l ∈ ℤ}. Thus, all paths I_l belonging to 𝓑 are just the arithmetic sequences (x_i|i ≤ n) of integers where the difference equals 1 and x₀ = ln if l is odd and the difference equals 1 and x₀ = (l + 1)n if l is even. Note that each element z ∈ 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 ∈ 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 closure space (ℤ^m, f₁(𝓑^m)) coincides with the m-dimensional Khalimsky space for every positive integer m. A digital Jordan curve theorem for the Khalimsky plane (ℤ², f₁(𝓑²)) was proved in [4] and a digital Jordan surface theorem for the Khalimsky space (ℤ³, f₁(𝓑³)) was proved in [13]. In [16], a Jordan curve theorem for the closure space (ℤ², f_n(𝓑²)) is proved (for an arbitrary integer n > 1). In the present note, we will focus on proving a digital Jordan surface theorem for the closure space (ℤ³, f₂(𝓑³)). Note that the graphs H² and H³ are simply the well known 8- and 26-adjacency graphs, respectively - cf. [11]. The path set 𝓑² ⊆ 𝓟₂(H²) is demonstrated in Figure 1 where the paths belonging to 𝓑² are marked by line segments directed from the first to the last terms of the paths.

$Figure 1 The path set 𝓑2 ⊆ 𝓟2(H2).$

Figure 1

The path set 𝓑² ⊆ 𝓟₂(H²).

From now on, we assume that n = 2. Hence, 𝓑 is a set of paths of length 2 (𝓑 ⊆ 𝓟₂(H)) and so is 𝓑³(𝓑³ ⊆ 𝓟₂(H³)). By [16], (ℤ³, f₂(𝓑³)) is connected.

Definition 3.1

Each of the following subsets of ℤ³ will be called a fundamental rectangle:

{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, z = 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, y = 4k, 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; x = 4k, 4l ≤ y ≤ 4l + 4, 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, y = x – 4k + 4l, 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, y = 4k + 4l + 4 – x, 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, z = x – 4k + 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, z = 4k + 4m + 4 – x}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, z = y – 4l + 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, z = 4l + 4m + 4 – y}, k, l, m ∈ ℤ.

Clearly, a subset T ⊆ ℤ³ is a fundamental rectangle if and only if there is a digital cube {(x, y, z); 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ, such that T is a (digital) face of the cube or is the intersection of the cube with the (digital) plane that is perpendicular to a face of the cube and contains one of the two (digital) diagonals of the face. Hence, every fundamental rectangle T consists of 25 points and has the form of a digital square parallel to a coordinate plane (hence perpendicular to the other two coordinate planes) or a digital rectangle perpendicular to a coordinate plane with the angle π4 between T and each of the other two coordinate planes. Thus, it is clear which sets of points (digital line segments) are the sides of a fundamental rectangle. By the help of Figure 1 (where the path set 𝓑² demonstrated is a two-dimensional projection of 𝓑³), we may easily see that every two different points of a fundamental rectangle may be joined by a 𝓑³-path contained in the rectangle. Thus, by Proposition 2.1, every fundamental rectangle is connected in (ℤ³, f₂(𝓑³)).

Definition 3.2

Each of the following subsets of ℤ³ will be called a fundamental triangle:

{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ x – 4k + 4l, z = 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, x – 4k + 4l ≤ y ≤ 4l + 4, z = 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4k + 4l + 4 – x, z = 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4k + 4l + 4 – x ≤ y ≤ 4l + 4, z = 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, y = 4l, 4m ≤ z ≤ x – 4k + 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, y = 4l, x – 4k + 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, y = 4l, 4m ≤ z ≤ 4k + 4m + 4 – x}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, y = 4l, 4k + 4m + 4 – x ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; x = 4k, 4l ≤ y ≤ 4l + 4, 4m ≤ z ≤ y – 4l + 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; x = 4k, 4l ≤ y ≤ 4l + 4, y – 4l + 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; x = 4k, 4l ≤ y ≤ 4l + 4, 4m ≤ z ≤ 4l + 4m + 4 – y}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; x = 4k, 4l ≤ y ≤ 4l + 4 4l + 4m + 4 – y ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ.

Clearly, a subset T ⊆ ℤ³ is a fundamental triangle if and only if there is a fundamental rectangle parallel to a coordinate plane, hence a square, such that T is one of the four (digital) half-square triangles obtained from the square (by splitting it along a diagonal). Thus, every fundamental triangle has the form of a (digital) isosceles right-angled triangle parallel to a coordinate plane. It is, therefore, clear which sets of points (digital line segments) are the sides of a fundamental triangle. As in the case of fundamental rectangles, fundamental triangles, too, may easily be seen, by the help of Figure 1 and Proposition 2.1, to be connected in (ℤ³, f₂(𝓑³)).

Definition 3.3

Each of the following subsets of ℤ³ will be called a fundamental prism:

{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ x – 4k + 4l, 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, x – 4k + 4l ≤ y ≤ 4l + 4, 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4k + 4l + 4 – x, 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4k + 4l + 4 – x ≤ y ≤ 4l + 4, 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, 4m ≤ z ≤ x – 4k + 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, x – 4k + 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, 4m ≤ z ≤ 4k + 4m + 4 – x}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4 4k + 4m + 4 – x ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, 4m ≤ z ≤ y – 4l + 4m}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, y – 4l + 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, 4m ≤ z ≤ 4l + 4m + 4 – y}, k, l, m ∈ ℤ,
{(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4 4l + 4m + 4 – y ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ.

Clearly, every fundamental prism has the form of a digital triangular prism with 75 points such that each of its faces is a fundamental triangle or a fundamental rectangle. Every cube {(x, y, z) ∈ 𝓩³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ 4l + 4, 4m ≤ z ≤ 4m + 4}, k, l, m ∈ ℤ, is the union of two different fundamental prisms having a face in common (and all fundamental prisms are obtained in this way).

Lemma 3.4

Every fundamental prism is connected in (ℤ³, f₂(𝓑³)) and so is every subset of ℤ³ obtained from a fundamental prism by removing some of its faces.

Proof

Let P be the fundamental prism given by condition (1) in Definition 3.3, i.e., P = {(x, y, z) ∈ ℤ³; 4k ≤ x ≤ 4k + 4, 4l ≤ y ≤ x – 4k + 4l, 4m ≤ z ≤ 4m + 4} where k = l = m = 0. Clearly, P is the union of the five triangles obtained as the intersection of P with the digital plane (parallel to the coordinate plane xy) {(x, y, z) ∈ ℤ³; z = k} where k is one of the integers 0, 1, 2, 3, 4. The projection of each of the five triangles onto the coordinate plane xy is the fundamental triangle (0, 0)(4, 0)(4, 4) in Figure 1 (which is connected in (ℤ², f₂(𝓑²)) by Proposition 2.1). Hence, each of the five triangles is connected in (ℤ³, f₂(𝓑³)). The same is true for every set obtained from any of the five triangles by removing some of its edges. Put C = ((3, 1, 0), (3, 1, 1), (3, 1, 2), (3, 1, 3), (3, 1, 4)). Since both ((3, 1, 2), (3, 1, 1), (3, 1, 0)) and ((3, 1, 2), (3, 1, 3), (3, 1, 4)) belong to 𝓑³, C is an 𝓑³-path. Thus, by Proposition 2.1, C is a connected subset of (ℤ³, f₂(𝓑³)). It is evident that C meets each of the five triangles and is contained in their union and the same is true even if some sides of the triangles are removed. Therefore, the fundamental prism P (which is the union of the triangles) is connected and the same is true if some faces of the prism are removed. If some of the integers k, l, m differ from 0, the proof is much the same. Thus, the Lemma is proved for fundamental prisms of the form (1) in Definition 3.3. For fundamental prisms of the other eleven forms, the proofs are done along similar lines.□

Theorem 3.5

(Digital Jordan Surface Theorem) Let S be a connected subset of (ℤ³, f₂(𝓑³)) which is the union of a finite (non-empty) set 𝓕 of fundamental triangles and fundamental rectangles such that

For every pair F₁, F₂ ∈ 𝓕, F₁ ≠ F₂, we have card(F₁ ∩ F₂) ≤ 1 or F₁ ∩ F₂ = J where J is a common side of F₁ and F₂.
For every F₁ ∈ 𝓕, if J is a side of F₁, then there exists exactly one F₂ ∈ 𝓕, F₁ ≠ F₂, such that J is a a side of F₂.

Then, S separates (ℤ³, f₂(𝓑³)) into exactly two components and the union of S with each of them is connected.

Proof

Let S satisfy the conditions of the statement. Clearly, S is a polyhedral surface which is the union of all faces of a polyhedron T_F ⊆ ℤ³ consisting of fundamental prisms. More precisely, T_F may be expressed as the union of a (finite) sequence of pairwise different fundamental prisms such that any two of them are disjoint or meet in just one face in common and every prism in the sequence, except for the first one, has a face in common with at least one of its predecessors. However, the set T_I = (ℤ³ – T_F) ∪ S, too, may be written as the union of such an (infinite) sequence of fundamental prisms. By Lemma 3.4, T_F, T_F – S, T_I, and T_I – S are connected in (ℤ³, f₂(𝓑³)).

It is obvious that every 𝓑³-walk C = (z_i | i ≤ k), k > 0 an integer, joining a point of T_F – S with a point of T_I – S meets S (i.e., meets a fundamental prism face contained in S). Thus, by Proposition 2.1, the set ℤ³ – S = (T_F – S) ∪ (T_I – S) is not connected in (ℤ³, f₂(𝓑³)). Hence, T_F – S and T_I – S are components of the subspace ℤ³ – S of (ℤ³, f₂(𝓑³)), T_F – S finite and T_I – S infinite, with T_F and T_I connected in (ℤ³, f₂(𝓑³)).□

Conclusion. We have introduced a closure operator on the digital space ℤ³, f₂(𝓑³), which provides a connectedness that allows for a digital Jordan surface theorem (Theorem 3.5). The Jordan surfaces introduced, i.e., the surfaces S satisfying the assumptions of Theorem 3.5, have the advantage over the Jordan surfaces with respect to the Khalimsky topology proposed in [13] that the angle between a pair of fundamental rectangles belonging to S may be π4 . For example, the surface of the letter M demonstrated in Figure 2 (where only the points on the edges of the letter are marked) is a Jordan surface in (ℤ³, f₂(𝓑³)) but it is not a Jordan surface with respect to the Khalimsky topology f₁(𝓑³) in the sense of [13] because there are four pairs of fundamental rectangles that meet at an angle of π4 . Therefore, the closure operator f₂(𝓑³) gives a convenient structure on ℤ³ for the study of three-dimensional digital images providing more flexible digital Jordan surfaces than the Khalimsky topology.

$Figure 2 A Jordan surface in (ℤ3, f2(𝓑3)).$

Figure 2

A Jordan surface in (ℤ³, f₂(𝓑³)).

Acknowledgement

This work was supported by 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] Klette R., Rosenfeld A., Digital Geometry - Geometric Methods for Digital Picture Analysis, Elsevier, Singapore, 2006.Search in Google Scholar

[2] Kong T.Y., Rosenfeld A., Digital topology: Introduction and survey, Comput. Vision Graphics Image Process., 1989, 48, 357–393.10.1016/0734-189X(89)90147-3Search in Google Scholar

[3] Rosenfeld A., Picture Languages, Academic Press, New York, 1979.Search in Google Scholar

[4] Khalimsky E.D., Kopperman R., Meyer P.R., Computer graphics and connected topologies on finite ordered sets, Topology Appl., 1990, 36, 1–17.10.1016/0166-8641(90)90031-VSearch in Google Scholar

[5] Khalimsky E.D., Kopperman R., Meyer P.R., Boundaries in digital plane, J. Appl. Math. Stochast. Anal., 1990, 3, 27–55.10.1155/S1048953390000041Search in Google Scholar

[6] Kiselman Ch.O., Digital Jordan curve theorems, In: G. Borgefors at al. (Eds.), Discrete Geometry for Computer Imagery, Lect. Notes Comput. Sci. 1953, Springer, Berlin-Heidelberg, 2000, 46–56.Search in Google Scholar

[7] Kong T.Y, Kopperman R., Meyer P.R., A topological approach to digital topology, Amer. Math. Monthly, 1991, 98, 902–917.10.1080/00029890.1991.12000810Search in Google Scholar

[8] ŠSlapal J., Jordan curve theorems with respect to certain pretopologies on ℤ², In: S. Brlek at al. (Eds.), Discrete Geometry for Computer Imagery, Lect. Notes Comput. Sci. 5810, Springer, Berlin-Heidelberg, 2009, 252–262.Search in Google Scholar

[9] Brimkov V.E., Klette R., Border and surface tracing - theoretical foundations, IEEE Trans. Patt. Anal. Machine Intell., 2008, 30, 577–590.10.1109/TPAMI.2007.70725Search in Google Scholar

[10] Kong T.Y., Roscoe W., Continuous analogs of axiomatized digital surfaces, Comput. Vision Graphics Image Process., 1985, 29, 60–86.10.1016/S0734-189X(85)90151-3Search in Google Scholar

[11] Morgenthaler D.G., Rosenfeld A., Surfaces in three dimensional digital images, Information and Control, 1981, 28, 227–247.10.21236/ADA092075Search in Google Scholar

[12] Reed M., Rosenfeld A., Recognition of surfaces in three dimensional digital images, Information and Control, 1982, 53, 108–120.10.21236/ADA125608Search in Google Scholar

[13] Kopperman R., Meyer P.R., Wilson R.G., A Jordan surface theorem for three-dimensional digital spaces, Discrete Comput. Geom., 1991, 6, 155–161.10.1007/BF02574681Search in Google Scholar

[14] Melin E., Digital surfaces and boundaries in Khalimsky spaces, J. Math. Imaging and Vision, 2007, 28, 169–177.10.1007/s10851-007-0006-9Search in Google Scholar

[15] Šlapal J., Graphs with a path partition for structuring digital spaces, Inform. Sciences, 2013, 233, 305–312.10.1016/j.ins.2013.01.011Search in Google Scholar

[16] Šlapal J., Closure operators on graphs for modeling connectedness in digital spaces, Filomat, 2018, 32, 5011–5032.10.2298/FIL1814011SSearch in Google Scholar

[17] Šlapal J., Galois connections between sets of paths and closure operators in simple graphs, Open Math., 2018, 16, 1573–1581.10.1515/math-2018-0128Search in Google Scholar

[18] Harrary F., Graph Theory, Addison-Wesley Publ. Comp., Reading, Massachussets, Menlo Park, California, London, Don Mills, Ontario, 1969.10.21236/AD0705364Search in Google Scholar

[19] Sabidussi G., Graph multiplication, Math. Z., 1960, 72, 446–457.10.1007/BF01162967Search in Google Scholar

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

Received: 2019-05-12

Accepted: 2019-10-19

Published Online: 2019-11-19

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0121

Keywords for this article

simple graph; path; closure operator; connectedness; digital space; digital surface; Khalimsky topology; Jordan surface theorem

Creative Commons

BY 4.0

Path-induced closure operators on graphs for defining digital Jordan surfaces

Article

Abstract

1 Introduction

2 Closure operators induced by sets of paths

Proposition 2.1

3 The closure operator on ℤ3 induced by a set of paths of length 2

Definition 3.1

Definition 3.2

Definition 3.3

Lemma 3.4

Proof

Theorem 3.5

Proof

Acknowledgement

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

3 The closure operator on ℤ³ induced by a set of paths of length 2