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 Z^{2} 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 Z^{2} 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 Z^{2} 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., (*x*_{0}, *x*_{1}, ..., *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, *x*_{0} = *x*_{n}, and (*x*_{i}| i < *n*) is a path. Given graphs *G*_{1} = (*V*_{1}, *E*_{1}, ) and *G*_{2} = (*V*_{2}, *E*_{2}), we say that *G*_{1} is a *subgraph* of *G*_{2} if *V*_{1} *⊆ V*_{2} and *E*_{1} *⊆ E* _{2}. If, moreover, *V*_{1} = *V*_{2}, then *G*_{1} is called a *factor* of *G*_{2}.

Recall [4] that the *strong product* of a pair of graphs *G*_{1} = (*V*_{1}, *E* _{1}, ) and *G*_{2} = (*V*_{2}, *E* _{2}, ) is the the graph *G*_{1⊗}*G*_{2} = (*V*_{1} × *V*_{2}, *E*) with the set of edges *E* = *{{*(*x*_{1}, *x*_{2}), (*y*_{1}, *y*_{2})*} ⊆ V*_{1} × *V*_{2}; 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 *G*_{1} and *G*_{2}, i.e., from the graph (*V*_{1} × *V*_{2}, *F*) where *F* = *{{*(*x*_{1}, *x*_{2}), (*y*_{1}, *y*_{2})*}*; *{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], *S*_{2}-closure operators and *S*_{2}-closure spaces are called *quasi-discrete*. *S*_{2}-topologies (*S*_{2}-topological spaces) are usually called *Alexandroff topologies* (*Alexandroff spaces*) - see [7]. Every *S*_{2}-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 *S*_{2}-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].

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