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

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 ℤm (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.


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 Z 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 de ne digital Jordan curves. In the classical approach to this problem, a pair of adjacency relations (4-and 8-adjacency) on Z 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 Z 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 Z 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 Z -see [9][10][11][12]. The topological approach employing the Khalimsky topology on Z 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 Z . 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 = , we get the well-known Khalimsky topology on Z m . We focus on the case of n = m = and, for the obtained closure operator on Z , 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 ( nite) sequence (x i | i ≤ n), i.e., (x , x , ..., xn), of pairwise di erent vertices of V such that x i is adjacent to x i+ whenever i < n. If, moreover, the members of ( The non-negative integer n is called the length of the walk (path) ( For the graph-theoretic background, we refer to [18].
Given graphs G j = (V j , E j , ), j = , , ..., m (m > an integer), we de ne their strong product to be the graph m j= G j = ( m j= V j , E) with the set of edges E = {{(x , x , ..., xm), (y , y , ..., ym)}; there exists a nonempty subset J ⊆ { , , ..., m} such that {x j , y j } ∈ E j for every j ∈ J and x j = y j for every j ∈ { , , ..., m} − J}. Note that the strong product di ers from the cartesian product of G j , j = , , ..., m, i.e., from the graph ( m The strong product of a pair of graphs coincides with that introduced in [19]. 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 (i) grounded (i.e., u∅ = ∅), The pair (X, u) is then called a closure space.
A closure operator u on X that is (iv) additive (i.e., u(A ∪ B) = uA ∪ uB whenever A, B ⊆ X) and (v) 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 > , a closure operator u on a set X and the closure space (X, u) are called an Smclosure operator and an Sm-closure space (brie y, an Sm-space), respectively, if the following condition is satis ed: S -topologies (S -topological spaces) are usually called Alexandro topologies (Alexandro 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 ( nite or in nite) 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.

Closure operators induced by 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 set of paths (path set for short) B ⊆ Pn(G), we put Let G j be a graph and B j ⊆ Pn(G j ) for every j = , , ..., m (m > an integer). Then, we put m m j= B j will be called the strong product of B j , j = , , ..., 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 = , , ..., m, we write B m instead of m j= B j . Let G be a graph with the vertex set V and B ⊆ Pn(G). For every X ⊆ V, we put It may easily be seen that fn(B) is an S n+ -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 , fn(B)). For the properties of the closure operators fn(B) see [17].
Recall [16] that, given a graph G = (V , E) and B ⊆ Pn(G), a sequence C = ( 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} ⊆ P (G).
We will need the following statement proved in [16]:

The closure operator on Z induced by a set of paths of length 2
Recall that the 2-adjacency graph (on Z) is the graph For every l ∈ Z, we put In the sequel, (for a given integer n > ) B will denote the set B ⊆ Pn(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 di erence equals and x = ln if l is odd and the di erence equals and x = (l + )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 (di erent) paths from B if and only if there is l ∈ Z with z = ln (in which case, z is the rst member of each of the paths I l and I l− 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 , f (B m )) coincides with the m-dimensional Khalimsky space for every positive integer m. A digital Jordan curve theorem for the Khalimsky plane (Z , f (B )) was proved in [4] and a digital Jordan surface theorem for the Khalimsky space (Z , f (B )) was proved in [13]. In [16], a Jordan curve theorem for the closure space (Z , fn(B )) is proved (for an arbitrary integer n > ). In the present note, we will focus on proving a digital Jordan surface theorem for the closure space (Z , f (B )). Note that the graphs H and H are simply the well known 8-and 26-adjacency graphs, respectively -cf. [11]. The path set B ⊆ P (H ) is demonstrated in Figure 1 where the paths belonging to B are marked by line segments directed from the rst to the last terms of the paths. From now on, we assume that n = . Hence, B is a set of paths of length 2 (B ⊆ P (H)) and so is B (B ⊆ P (H )). By [16], (Z , f (B )) is connected.
De nition 3.1. Each of the following subsets of Z will be called a fundamental rectangle: Clearly, a subset T ⊆ Z is a fundamental rectangle if and only if there is a digital cube {(x, y, z); k ≤ x ≤ k + , l ≤ y ≤ l + , m ≤ z ≤ m + }, 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 π 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 B demonstrated is a two-dimensional projection of B ), we may easily see that every two di erent points of a fundamental rectangle may be joined by a B -path contained in the rectangle. Thus, by Proposition 2.1, every fundamental rectangle is connected in (Z , f (B )).

De nition 3.2.
Each of the following subsets of Z will be called a fundamental triangle:

Conclusion.
We have introduced a closure operator on the digital space Z , f (B ), 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 π . 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 (Z , f (B )) but it is not a Jordan surface with respect to the Khalimsky topology f (B ) in the sense of [13] because there are four pairs of fundamental rectangles that meet at an angle of π . Therefore, the closure operator f (B ) gives a convenient structure on Z for the study of three-dimensional digital images providing more exible digital Jordan surfaces than the Khalimsky topology.