Abstract
Covering is a type of widespread data representation while covering-based rough sets provide an efficient and systematic theory to deal with this type of data. Matroids are based on linear algebra and graph theory and have a variety of applications in many fields. In this paper, we construct two types of covering cycle matroids by a covering and then study the graphical representations of these two types of matriods. First, through defining a cycle graph by a set, the type-1 covering cycle matroid is constructed by a covering. By a dual graph of the cycle graph, the covering can also induce the type-2 covering cycle matroid. Second, some characteristics of these two types of matroids are formulated by a covering, such as independent sets, bases, circuits, and support sets. Third, a coarse covering of a covering is defined to study the graphical representation of the type-1 covering cycle matroid. We prove that the type-1 covering cycle matroid is graphic while the type-2 covering cycle matroid is not always a graphic matroid. Finally, relationships between these two types of matroids and the function matroid are studied. In a word, borrowing from matroids, this work presents an interesting view, graph, to investigate covering-based rough sets.
1. Introduction
Covering is a type of common and important data organization mode, and it most appears in incomplete information/decision systems based on symbolic data [1, 2], numeric and fuzzy data [3, 4]. Covering-based rough set theory [5, 6] is an efficient tool to process these types of data. Recently, this theory has attracted much research interest with fruitful achievements on both theory and applications. For example, it has been applied to build axiomatic systems [7, 8] and establish knowledge reduction approaches [9, 10]. Moreover, it also has been used to construct covering structures [11, 12] and define minimal covering reducts [13, 14]. However, this theory has its own limitation in dealing with some hard problems including knowledge reduction. In order to improve its ability to process those hard problems, some other mathematical theories, such as fuzzy set theory [15, 16], topology [5, 17], Boolean algebra [18, 19], and matroid [20, 21] have been combined with covering-based rough set theory.
Matroid theory [22] proposed by Whitney is a generalization of linear algebra, graph theory, and transcendence theory. The original purpose of this theory is to formalize the similarities between the ideas of independence and rank in graph theory and those of linear independence and dimension in the study of vector spaces [23]. Matroids have been applied to a number of fields, such as combinatorial optimization [24], algorithm design [25], and information coding [26]. Matroids can provide well-established platforms for greedy algorithms, which may help to process those problems that are difficultly solved by rough sets. Thus, several matroidal structures of rough sets have been studied from different viewpoints, such as binary relations [27], coverings [28, 29], and graphs [30]. Therefore, it makes sense to study matroidal structures of coverings through graphs.
In this paper, inspired by union of matroids and cycle matroid, we construct two types of covering cycle matroids by a covering and then study the graphical representations of these two types of matroids. By defining a cycle graph through a set, any block of the covering can induce a cycle graph and a dual graph of the cycle graph, and then some cycle matroids and their dual matroids are obtained by these cycle graphs and dual graphs, respectively. Therefore, type-1 covering cycle matroid is obtained by the union of these cycle matroids, and type-2 covering cycle matroid is obtained by the union of the dual matroids of these cycle matroids. The independent sets, bases, and support sets of these two types of matroids are represented by the covering. In particular, the independent sets of the type-1 covering cycle matroid are equivalently represented by the lower approximations, and the support sets of the type-2 covering cycle matroid are equivalently represented by the upper approximations. We prove that the type-1 covering cycle matroid induced by a covering is equal to the one induced by the intersection reduct of the covering. We also investigate the graphical representations of these two types of matroids. By redefining a covering into a coarse covering, the matroid induced by a graph which is generated by the coarse covering is equal to the type-1 covering cycle matroid induced by the original covering. As for the type-2 covering cycle matroid, it is not always a graphic matroid. We also study relationships between these two types of matroids and the function matroid. Results show that these two types of covering cycle matroids are not always dual, and these three kinds of matroids are equal when the cardinality of any block of a covering is equal to 2.
The rest of this paper is organized as follows. Section 2 reviews some fundamental definitions about covering-based rough sets, matroids, and graphs. In Section 3, we construct two types of covering cycle matroids by a covering and then study the graphical representations of these two types of matroids. Section 4 studies relationships between three kinds of matroids, which are induced by a covering, respectively. Finally, this paper is concluded in Section 5.
2. Basic Definitions
This section recalls some fundamental definitions related to covering-based rough sets, matroids, and graphs.
2.1. Covering-Based Rough Sets
Covering is a common type of data structure, and it can characterize the practical problems with extensive coverage.
Definition 1 (covering [31]). Let be a universe of discourse and a family of nonempty subsets of . If , then is called a covering of .
As we know, a partition of is certainly a covering of ; so the concept of a covering is an extension to the concept of a partition.
Neighborhoods are important concepts in rough sets, and they can describe the maximal dependence to an object.
Definition 2 (indiscernible neighborhood [6]). Let be a covering of and . is called the indiscernible neighborhood of with respect to . When there is no confusion, we omit the subscript .
In knowledge discovery, each element in a covering is called knowledge. As we know, some knowledge may be redundant. That is to say, removing those redundant knowledge cannot change the approximation accuracy. To deal with those redundant knowledge, a notion of reducible element is proposed and it has many different forms. For example, the reducible element proposed by Zhu and Wang [8] is different from the one defined by Y. Yao and B. Yao [12].
Definition 3 (see [12]). Let be a covering of . If is an intersection of some elements in , then is said to be an intersection reducible element in , otherwise is said to be an intersection irreducible element. If every element of is an intersection irreducible element, then is said to be intersection irreducible; otherwise is said to be intersection reducible.
Theorem 4 (see [12]). Let be a covering of . Suppose is an intersection reducible element of , then, for all , is an intersection reducible element of if and only if is an intersection reducible element of .
We can simplify a covering by iteratively removing reducible elements to obtain reduced forms of .
Definition 5 (see [12]). Let be a covering of . If is the set of all intersection reducible elements of , the set is called the intersection reduct of and denoted by .
For a covering of , if is an intersection reducible element in , then is still a covering of . Therefore, is a covering of .
In covering-based rough sets, an object is described by a pair of approximations. In the following definition, we introduce a pair of widely used approximations.
Definition 6 (approximations [6]). Let be a covering of . For all , are called the lower and upper approximations of , respectively.
2.2. Matroids
Matroids are algebraic structures that capture and generalize linear independence in vector spaces. A characteristic of matroids is that they are defined in many different but equivalent ways. In the following, we introduce one defined by independent sets.
Definition 7 (matroid [22]). A matroid is a pair where is a finite set, and (independent sets) is a family of subsets of satisfying the following three conditions:; if , and , then ; if , , and , then there exists such that , where denotes the cardinality of .
For a better understanding of the different definitions of a matroid, some operations will be firstly introduced as follows.
Definition 8 (see [22]). Let be a family of subsets of . One can denote , , , , , , .
In order to introduce the dual matroid of a matroid, we first recall the definition of a base in a matroid. Any base of a matroid generalizes the maximal linearly independent vector group of a vector space and the spanning tree of a graph.
Definition 9 (base [22]). Let be a matroid. A maximal independent set of is called a base of , and the set of all bases of is denoted by , that is, .
Clearly, when the set of all independent sets of a matroid is given, one can determine and vice versa. By the bases in matroids, we introduce the concept of the dual matroid of a matroid, which is an extension of the orthogonal complement space of a vector space.
Definition 10 (dual matroid [22]). Let be a matroid and , where denotes the complement of in . Then is the family of all bases of a matroid which is called the dual matroid of and denoted by .
The complement of the independent sets in power sets is dependent ones. And a minimal set of the dependent sets is called a circuit of the matroid. A matroid uniquely determines its circuits and vice versa.
Definition 11 (circuit [22]). Let , be a matroid. A minimal dependent set in is called a circuit of , and we denote the family of all circuits of by , that is, .
Matroids have many equivalent definitions. Support sets can uniquely determine one matroid. Support sets are defined as follows.
Definition 12 (support set [22]). Let , be a matroid. For all , if there exists a base such that , then is called a support set of , and we denote the family of all support sets of by .
From the viewpoint of circuits, matroids are viewed as a generalization of graphs. In the following, we will recall the definition of cycle matroid.
Proposition 13 (cycle matroid [22]). Let be a graph. Denote (as a subgraph) does not contain cycles}. Then is a matroid, and it is called the cycle matroid of and denoted as .
Union of matroids was introduced by Nash-Williams in 1966. In the following, we will recall the definition of union of matroids on the same universe.
Definition 14 (union of matroids [22]). Let , , and be a group of matroids on the universe . Then is a matroid, where , , which is called the union of , and and denoted by .
The graphical representation of matroids is an important content in matroids. For a matroid , is a graphic matroid [22] if there exists a graph such that . An equivalent characterization of a graphic matroid is given in the following.
Theorem 15 (see [22]). A matroid is graphic if and only if it has no minor which is one of the , , , , and .
A minor of a matroid is another matroid that is obtained from by a sequence of restriction and contraction operations. We will introduce two special matroids in the following two definitions.
Definition 16 (restriction matroid [22]). Let , be a matroid. For any , we define . Then there exists a matroid on with as its independent sets, and is called the restriction matroid of on .
Definition 17 (uniform matroid [22]). Let . For an integer , we define . Then , forms a matroid, and it is called a uniform matroid and denoted by .
2.3. Graphs
Graph theory provides an intuitive way to interpret and comprehend a number of practical and theoretical problems. Theoretically, a graph is an ordered pair consisting of vertices and edges that connect these vertices.
A graph [32] is a pair , consisting of a set of vertices and a set of edges such that . A path [32] is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the list. A cycle [32] is a graph with an equal number of vertices and edges whose vertices can be placed around a circle so that two vertices are adjacent if and only they appear consecutively along the circle. A loop [32] is an edge whose endpoints are equal.
3. Two kinds of Matroids Induced by a Covering
In this section, we define a cycle graph by a set and then construct two kinds of matroidal structures by a covering.
3.1. Type-1 Covering Cycle Matroid
In this section, we construct the type-1 covering cycle matroid by a covering and then study the graphical representation of the matroid. The relationship between the matroid and a matroid induced by a graph generated by the covering is also studied.
In order to establish the connection between coverings and matroids, we first propose a notion as follows.
Definition 18 (cycle graph induced by a set). Let and . We define a graph induced by as follows:(1);(2) where the path is a cycle.
As we know, a set is an unordered collection with no duplicate elements. That is to say a set can induce different cycle graphs. An example to illustrate this feature is given in the following.
Example 19. Let , , , , and the cycle graphs induced by . As shown in Figure 1, and are represented by and , respectively.
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Given a set, according to Definition 18, the set can induce a cycle graph. Given a graph, according to Proposition 13, the graph can induce a cycle matroid. Although a set can induce some different cycle graphs, all the cycle matroids induced by these graphs have only one circuit which is equal to the set. Hence the cycle matroids induced by these graphs are the same, and any maximal proper subset of the given set is a base of the cycle matroid generated by a cycle graph which is induced by the given set.
Proposition 20. Let be a subset of , a cycle graph induced by , and the cycle matroid induced by . Therefore .
Proof. According to Definition 18, the path is a cycle. Hence for any , does not contain cycles. According to Proposition 13, (as a subgraph) does not contain cycles.
The following example illustrates the uniqueness of the cycle matroid generated by different cycle graphs induced by the same set.
Example 21 (continued from Example 19). Let , be two different cycle graphs induced by and and the cycle matroids induced by and , respectively. By direct computation, and . Moreover, . Hence .
Given a covering of a universe, according to Definition 18, every block of the covering can induce a cycle graph. Hence any block of a covering can induce a matroid. What we concern about is that if the covering can generate a matroid. According to Definition 14, we can obtain a matroid by the union of the matroids, which are induced by a block of the given covering, respectively.
Proposition 22. Let be a covering of , a cycle graph induced by , and the cycle matroid induced by . Therefore is a matroid and it is denoted by , where .
Proof. For any , we define a matroid on as , where and is a circuit of . According to Definition 14, it is straightforward that . So is a martoid.
Definition 23 (type-1 covering cycle matroid). Let be a covering of . is called type-1 covering cycle matroid induced by .
Example 24. Let and , where , , and . As shown in Figure 2, three cycle graphs , , and induced by , , and are represented by Figures 2(a), 2(b), and 2(c), respectively. The cycle matroids induced by , , and are , and . By direct computation, , , and . Therefore the type-1 covering cycle matroid induced by is , where .
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For a set, a maximal independent set of the cycle matroid generated by a cycle graph induced by the set is a maximal proper subset of the set. In the following proposition, we use all the blocks of a covering to represent a base of the type-1 covering cycle matroid induced by the covering.
Proposition 25. Let be a covering of and the type-1 covering cycle matroid induced by . Then .
Proof. According to Definition 9, . Since is a cycle graph induced by and the cycle matroid induced by , according Proposition 20, . Hence .
In the following proposition, when a covering degenerates a partition, we represent the bases of the type-1 covering cycle matroid induced by the covering.
Proposition 26. Let be a covering of and the type-1 covering cycle matroid induced by . If is a partition of , then .
Proof. Since is a partition of and for any , for any , . According to Proposition 25, .
An equivalent formulation of the independent sets of the type-1 covering cycle matroid induced by a partition is provided from the viewpoint of lower approximations. In fact, a subset of a universe is an independent set if and only if the lower approximation of the subset is equal to empty set.
Proposition 27. Let be a covering of and the type-1 covering cycle matroid induced by . If is a partition of , then .
Proof. For any , there exists such that . Since is a partition of , there exists such that , and for any . So for any . Therefore, for any , that is, . Conversely, for any , if , then for all . So . According to Proposition 20, there exists such that . Hence . Moreover, . So , that is, . To sum up, this completes the proof.
When a covering of a universe is not a partition of the universe, there exists an independent set whose lower approximation is not empty. Example 24 can be used to illustrate this feature. Since , . But .
A covering can induce the type-1 covering cycle matroid. We would like to know whether there exist two different coverings such that the type-1 covering cycle matroids induced by them are equal. As shown in the following proposition, the type-1 covering cycle matroid induced by any covering is equal to the one induced by the covering , if is an intersection reducible element of .
Proposition 28. Let be a covering of . If is an intersection reducible element of , then .
Proof. Since is an intersection reducible element of , there exist some elements in such that . For any , . If , then . Hence . If , then for any and . Since , . So . Hence , . Therefore, .
The type-1 covering cycle matroid induced by any covering is equal to the one induced by the intersection reduct of the covering.
Corollary 29. Let be a covering of . .
Proof. Suppose are all intersection reducible elements of , then according to Proposition 28, . According to Theorem 4, is also an intersection reducible element of the covering . Hence . Similarly, we can prove that . Therefore, .
In the rest of this subsection, the main task is to study the graphical representation of the type-1 covering cycle matroid. First, we redefine a given covering of a universe into another covering of the universe.
Definition 30. Let be a covering of . A family of subsets of is defined as follows: is then also a covering of and it is called the coarse covering of . When , is the indiscernible neighborhood of with respect to covering .
Example 31. Let and , where , , , , and . By direct computation , , . So .
Proposition 32. Let be a covering of and the coarse covering of . is a partition of .
Proof. For any , , , . If , then we need to prove that . If there exists such that , then for any , , and . So . Therefore, , that is, , which is contradictory to . So . Similarly, we can prove that . Consequently, , that is, is a partition of .
Relationships between those blocks of a covering and any block in the coarse covering of the covering are shown in the following two propositions.
Proposition 33. Let be a covering of and the coarse covering of . For any and , if , then .
Proof. For any and , . So .
Proposition 34. Let be a covering of and the coarse covering of . For any , if and for any and , then .
Proof. Since is a covering of and , . According to Proposition 33, . So .
As we know, matroids and graphs coincide with each other when the circuit of a matroid is degenerated to the cycle of a graph. The following proposition shows relationships between each block in the coarse covering of a given covering and circuits of the type-1 covering cycle matroid induced by the given covering.
Proposition 35. Let be a covering of and the coarse covering of . For any , if , then .
Proof. Since , . If there exists such that , then for any . For any , if , then we need to prove that for any . According to Proposition 34, there exists such that . For any , there exists such that and . So . Since , according to Proposition 32, . If for any and , , then , which is contradictory to . So there exists with the condition that for any , and , there exists such that and . Hence for any , . Similarly, we can prove that there exists with the condition that, for any , and , there exists such that and . And for any , . Since is a finite set, there exist a finite set with for any and an element such that . Therefore, for any . So , that is, .
In the following proposition, we study relationships between the coarse covering of a given covering and circuits of the type-1 covering cycle matroid induced by the given covering.
Proposition 36. Let be a covering of and the coarse covering of . For any , , if and , then .
Proof. For any , , according to Proposition 35, . Conversely, for any , . Hence , that is, . There exists an index set such that for any and , and . So . Suppose there exists a nonempty index set with the condition that, for any and , and there exists such that . Since , . There exists a finite set with for any such that and for any , . For any , and , and . If , then , which is contradictory to . So , that is, , which is contradictory to . Therefore , that is, . Since for any , there exists such that for any and , that is, . Therefore, .
In the following definition, we introduce a method to get a connected graph by some disjoint graphs.
Definition 37 (vertex identification [22]). Suppose that the graph is obtained from the disjoint graphs and by identifying the vertices of and of as the vertex of . This operation is called a vertex identification.
A connected graph is constructed by some disjoint graphs through vertex identification. These disjoint graphs are, respectively, induced by a block in the coarse covering of a covering.
Definition 38. Let be the coarse covering of , a cycle graph induced by , and a path induced by , where , and . We define a graph by vertex identification of , and , and this graph is denoted by . We say is a graph induced by .
The following example is about the operation of vertex identification and a graph obtained by the coarse covering of a covering.
Example 39 (continued from Example 31). . Since , and , is a path and both and are cycle graphs. Those three graphs are shown in Figure 3. If is a graph obtained by vertex identification of these three graphs, then is a graph induced by . is shown in Figure 4.
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It is obvious that there are some different graphs which can be obtained by the coarse covering of a covering through the operation of vertex identification. Cycles of each graph are the same despite these graphs are different. And these cycles are those blocks in the coarse covering which do not belong to the family of all independent sets of the type-1 covering cycle matroid induced by the original covering. From the viewpoint of the circuit of a matroid, a matroid induced by a connected graph constructed from the coarse covering of a given covering is equal to the type-1 covering cycle matroid induced by the given covering.
Theorem 40. Let be a covering of , the coarse covering of , and a graph induced by . Then .
Proof. We need to prove only that . Since is the graph obtained by vertex identification of , then , where is a family of subsets of and it has a property that for any . According to Proposition 36, . Therefore, .
3.2. Type-2 Covering Cycle Matroid
In this section, type-2 covering cycle matroid is defined, and then the graphical representation of this type of covering cycle matroid is studied.
Every connected plane graph has a natural dual graph such that . The dual is formed by associating a vertex of with each face of and including a dual edge in for each edge of , such that the endpoints of the edge are the vertices for the faces on the two sides of .
In graph theory, a plane graph is a graph in which no edges cross each other. For any set , a cycle graph induced by has no edges cross each other, so the graph is a plane graph and has a dual graph. Since has two faces, its dual graph has only two vertices. In order to further understand the notion about a dual graph of a plane graph, an example is given in the following.
Example 41 (continued from Example 24). As shown in Figure 5, the dual graphs of the cycle graphs , , and are represented by Figures 5(a), 5(b), and 5(c), respectively.
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Given a set, there are some different cycle graphs induced by the set, but these cycle graphs have the same dual graph. As we know that any edge in any cycle graph induced by the set has two faces, so all the dual graphs of these cycle graphs have only two vertices and any two edges of each dual graph form a cycle. That is to say, any two edges of each dual graph are adjacent. So these dual graphs are the same. When the set is a singleton, the cycle graph induced by the set is a loop. Since the edge of a loop has two faces, the dual graph of the loop has two vertices and an edge. That is to say, a singleton whose element is the edge of the dual graph of the loop is always a base of the matroid induced by this dual graph. Therefore, for a cycle graph, a singleton that consists of any edge of the dual graph of the cycle graph is a base of the matroid induced by the dual graph. Then the following proposition can be obtained.
Proposition 42. Let be a subset of , the dual graph of a cycle graph induced by , and the cycle matroid induced by . Then .
Proof. According to Proposition 13, Definitions 18 and 10, Proposition 20, and the notion of a dual graph of a plane graph, it is straightforward.
The above proposition shows that a singleton consists of any element in the given set that is an independent set of the dual matroid of the cycle matroid, which is induced by a cycle graph. And this cycle graph is induced by the given set. According to Definition 14, the following proposition can be obtained easily.
Proposition 43. Let be a covering of , the dual graph of a cycle graph induced by , and the cycle matroid induced by . Then is a matroid and it is denoted by , where .
Proof. For any , . So according to Proposition 42, . Hence , . For any , we define a matroid on as , where and is a circuit of . Moreover ; so, according to Definition 14, is the set of all independent sets of the matroid . Therefore ; that is, is a matroid.
Definition 44 (type-2 covering cycle matroid). Let be a covering of . Then is called type-2 covering cycle matroid inducd by .
Example 45 (continued from Example 41). The dual matroids of three cycle matroids induced by three cycle graphs generated by , , and are , and . By direct computation, , , . If , then .
The following proposition shows relationships between the set of all bases of the type-2 covering cycle matroid induced by a covering and those blocks of the covering.
Proposition 46. Let be a covering of . If is the type-2 covering cycle matroid induced by , then .
Proof. According to Proposition 43, . Since , .
When a covering is a partition of the universe, relationships between the set of all bases of the type-2 covering cycle matroid and those blocks of the covering are investigated in the following proposition.
Proposition 47. Let be a covering of and the type-2 covering cycle matroid induced by . If is a partition of , then .
Proof. According to Proposition 46, . Since is a partition of , for any , . Hence .
In the following proposition, we connect upper approximations with support sets of the type-2 covering cycle matroid. In fact, when a covering is a partition of a universe, a subset of the universe is a support set of the type-2 covering cycle matroid induced by the covering if and only if the upper approximation of the subset is equal to the universe.
Proposition 48. Let be a covering of and the type-2 covering cycle matroid induced by . If is a partition of , then .
Proof. For any , according to Definition 12, there exists such that . Since is a partition of , according to Proposition 47, . Hence for all . Since , , that is, for all . Therefore, , that is, . Conversely, for any , . Since is a partition of , for all ; that is, there exist such that . Hence . Since is a partition of , according to Proposition 47, . According to Definition 12, , that is, . To sum up, this completes the proof.
When a covering of a universe is not a partition of the universe, there exists a subset of the universe which is not the support set of such that the upper approximation of the subset is equal to the universe.
Example 49 (continued from Example 45). Since , there does not exist such that . So . But .
An equivalent formulation of the family of all circuits of the type-2 covering cycle matroid induced by a covering is provided by those blocks of the covering. In fact, when the covering is a partition of a universe, a subset of the universe is a circuit if and only if it is contained in a block of the covering and its cardinality is equal to 2.
Proposition 50. Let be a partition of and the type-2 covering cycle matroid induced by . Therefore and .
Proof. Since , for any . For any , . If there exists such that , then since is a partition of . If and , then and . So . So there exists such that and , that is, and . Conversely, for any and , there exists such that . Since is a partition of and , and for any . So , that is, . To sum up, this completes the proof.
A connected graph is constructed by the dual graphs of some cycle graphs through vertex identification. These cycle graphs are, respectively, induced by an element in a partition of a universe.
Definition 51. Let be a partition of and the dual graph of a cycle graph induced by . We define a graph by vertex identification of , and , and this graph is denoted by . We say is a graph induced by .
An example about the operation of vertex identification of the dual graphs of the cycle graphs, which are, respectively, induced by an element of a partition, is shown in the following.
Example 52. Let and , where , . As shown in Figure 6, the dual graphs of two cycle graphs and are represented by Figures 6(a) and 6(b), respectively. Figure 7 is the representation of which is obtained by vertex identification of and .
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Theorem 40 shows the type-1 covering cycle matroid induced by any covering is graphic and a graph corresponding to the matroid is constructed by the covering via an indirect route. In the following theorem, we will discuss the relationship between a connected graph induced by a partition and the type-2 covering cycle matroid induced by the partition. It is obvious that there are some different graphs obtained by the partition through the operation of vertex identification. The cycles of each graph are the same while these graphs are different.
Theorem 53. Let be a partition of , a graph obtained by vertex identification of , and the type-2 covering cycle matroid induced by . Then .
Proof. We need to prove only that . For , if and , then for any , is a edge of and is a cycle of . Therefore is a cycle of . If and , then . According to Proposition 50, . Therefore, .
For any covering of the universe, we want to know that whether the type-2 covering cycle matroid is graphic. A counterexample is given in the following.
Example 54. Let , and the type-2 covering cycle matroid induced by , where , , . By direct computation, , , , , , . If , then . So . Hence is not a graphic matroid.
4. Relationships between Three Kinds of Matroids
In this section, we study relationships between type-1 covering cycle matroid, type-2 covering cycle matroid, and function matroid.
For a covering, we can induce two types of covering cycle matroids. Naturally, we will consider whether these two types of covering cycle matroids are equal. The following theorem shows that the type-1 covering cycle matroid induced by a covering is equal to the type-2 covering cycle matroid induced by the covering when the cardinality of every block of the covering is equal to two.
Theorem 55. Let be a covering of . If for all , , then .
Proof. Since for all , , then according to Definition 18 and Proposition 13, for any . For any , .
According to Proposition 22, . Hence .
The upper approximation number provides a tool to quantify covering-based rough sets. The upper approximation number is defined as follows.
Definition 56 (see [28, 29, 33]). Let be a covering of . For all , is called the upper approximation number of with respect to . When there is no confusion, we omit the subscript .
In the following, a matroid is defined through the upper approximation number. We say that the matroid is the function matroid induced by the covering.
Definition 57 (see [28, 29]). Let be a covering of . Then we say a martoid where for all , which is called the function matroid induced by .
Given a covering, the upper approximation number of any base of the function matroid induced by the covering is equal to the cardinality of the covering.
Proposition 58. Let be a covering of , the function matroid induced by , and the family of all the bases of . For all , .
Proof. If , that is, there exists such that , then for any , . Since , . Hence for any . Therefore, for any , if , then . If , then there exists such that , and . So . Since , is not a base of , which is contradictory to . Therefore, for all .
The following theorem shows that the function matroid induced by a covering is equal to the type-2 covering cycle matroid induced by the covering.
Theorem 59. Let be a covering of . Then .
Proof. We need to prove only . Let be the bases of the function matroid . For any , there exists such that and . Hence ; that is, there exists such that and , that is, . According to in Definition 7, for any . Hence ; that is, . For any , . So . For any , . Hence . Therefore, for all , . So . To sum up, this completes the proof.
The following corollary shows that the type-1 covering cycle matroid induced by a covering is equal to the function matroid induced by the covering when the cardinality of every block of the covering is equal to two.
Corollary 60. Let be a covering of . If for all , , then .
Proof. According to Theorems 55 and 59, it is straightforward.
For a covering , maybe people want to ask a question about the relationship between and as follows: Could and be dual matroids? Next, we will answer this question.
Example 61 (continued from Examples 24 and 45). According to Examples 24 and 45, and . So for any . Therefore and are not dual matroids.
Example 61 shows type-1 and type-2 covering cycle matroids induced by the same covering are not always dual. The following theorem shows that when a covering of a universe is a partition of the universe, these two types of covering cycle matroids induced by the covering are dual with each other.
Theorem 62. If is a partition of , then .
Proof. According to Propositions 26 and 47 and Definition 10, it is straightforward.
5. Conclusions
In this paper, we constructed two types of covering cycle matroids by a covering and studied the graphical representations of these two types of matriods. Some concepts of these two types of matroids were studied by those blocks of the covering, such as independent sets, bases, circuits, and support sets. We proved that the type-1 covering cycle matroid is a graphic matroid while the type-2 covering cycle matroid is not always a graphic matroid. These results provide a platform for studying covering-based rough sets through matroidal approaches. With the advantage of matroids, covering cycle matroid will help to develop some efficient algorithms for processing the data organized by coverings. In future, we will use both matroid theory and graph theory simultaneously to study rough set theory.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China under Grant no. 61170128, the Natural Science Foundation of Fujian Province, China, under Grant nos. 2011J01374 and 2012J01294, and the Science and Technology Key Project of Fujian Province, China, under Grant no. 2012H0043.