Abstract
Polya’s plausible reasoning methods are crucial not only in discovery of mathematics results, modeling methods, and data processing methods but also in many practical problems’ solving. This paper exemplifies how to use Polya’s plausible reasoning methods to generalize the popularized notion of 2-connectedness of graphs to a more universal notion of the connected degree of fuzzifying matroids. We introduce the connectedness of fuzzifying matroids, which is generalized from 2-connectedness of graphs, connectedness of matroids, and 2-connectedness of fuzzy graphs. Moreover, the connected degree of fuzzifying matroids is presented by considering the fuzziness degree of connectedness. It is proved that a fuzzifying matroid is connected, which is equivalent to its connectedness degree is the biggest (i.e., ). This, together with other properties of the connected degree of fuzzifying matroids, demonstrates the rationality of the proposed notion. Finally, we describe the concepts of this paper through some examples.
1. Introduction
Whitney in his fundamental paper [1] defined a matroid as an abstract generalization of a graph and a matrix. One of the great beauties of the subject of the matroid theory is that there are so many equivalent descriptions of a matroid. Bases, circuits, rank function, or closure operator is also sufficient to uniquely determine the matroid besides independent sets [2, 3]. In addition, it is well known that matroids are of great significance in combinatorial optimization and are just the structure that can use the greedy algorithm to find the optimal solution of some problems [4, 5]. Combined with fuzzy set theory [6], some fuzzy concepts of a matroid are given. One is using a family of fuzzy sets instead of that of sets, a Goetschel–Voxman fuzzy matroid was proposed in [7] and was later on widely studied by the various researchers [8–16]. The other is using a mapping from to rather than a family of sets, a fuzzifying matroid was introduced by Shi, and it can also be determined uniquely by its fuzzifying rank function [17, 18]. Subsequently, some other equivalent descriptions of a fuzzifying matroid were discussed including base-map and circuit-map [19], three kinds of fuzzifying operators [20, 21], fuzzifying nullity [22]. Also, as its applications, a fuzzifying matroid is just the structure of the fuzzifying greedy algorithm for fuzzy optimization problems [23, 24]. Thus, Shi’s fuzzification method maintains matroids’ features.
We know that there is an important connection between graphs and matroids. A graph (graphs in this paper are always loopless and without isolated vertices and have at least three vertices). can induce a matroid which is called its cycle matroid and denoted by , where = {}. Motivated by analogies in graph theory, the connectedness of a matroid is defined. Here, is connected, which is equivalent to is 2-connected; hence, the connectedness for matroids corresponds directly to the idea of 2-connectedness for graphs. It is natural to ask that is there fuzzifying matroid concept that corresponds directly to the idea of connectedness, or 2-connectedness, for fuzzy graphs?
Our main aim is to give a concept of connectedness for fuzzifying matroids, which extends the corresponding notion for fuzzy graphs. We begin by recalling some basic notions and results to be used in this paper. In Section 3, by using Polya’s plausible reasoning method, we show the course of discovering the notion of connected degree of fuzzifying matroids. In Section 4, we generalize the concept of 2-connectedness from matroids to fuzzifying matroids, which corresponds directly to the idea 2-connectedness for fuzzy graphs. In Section 5, we consider the fuzziness degree of connectedness for fuzzifying matroids. Using the degree to which a set is a separator, the notion of connected degree for a fuzzifying matroid is introduced. We show that the relationship between the connected degree of a fuzzifying matroid and the connectedness of its corresponding level matroids and investigate the connected degree for the dual of a fuzzifying matroid. We close the section with various examples to illustrate the concepts of this paper.
2. Preliminaries
In this paper, we shall denote by a given finite set and by the set of all subsets of . Let and define = {}, = {}, = {}.
We begin with the definition of a matroid.
Definition 1. (cf. [2, 3]) A matroid is a finite set and a collection of subsets of (called independent sets) such that (I1)–(I3) are satisfied, denoted by . (I1) ; (I2) is a descending family; (I3) For any two elements with , one element can be found so that A {e} belongs to .It is important (for understanding the notion of fuzzifying matriod in the following Step 5) to notice that a matroid can also be defined equivalently as a finite set and a collection of subsets of (called circuits) satisfying three circuit axioms [2]: ; is an antichain; for any two different elements and , one element can be found to be included in − {e}. Specifically, take a matroid (i.e., which means that satisfies (I1)–(I3)), then, is the family of all circuits for it (i.e., satisfies (C1)–(C3)); conversely, take a matroid (i.e., satisfies (C1)–(C3)), then is the family of all independent sets for it (i.e., satisfies (I1)–(I3)) and .
Moreover, if is a matroid, its rank function is defined as a mapping from to the set of natural numbers by {} and can uniquely determine the matroid [3].
Next, we present some other notations, notions, and results needed in this paper. As usual, denotes the collection of fuzzy subsets on , where a fuzzy subset on is a mapping from to . If and , we will often denote the set of satisfying and the set of satisfying by and , respectively. Fuzzy natural numbers and their operations have been introduced [17, 22, 23, 25]. An antitone mapping from the set of natural numbers to is called a fuzzy natural number if and {} = 0. We will use to denote the collection of fuzzy natural numbers. Let be fuzzy natural numbers, their sum, written is also a fuzzy natural numbers such that for every , {}, their subtraction is given by {}. If we regard a natural number as a fuzzy natural number such that for and for , then, we can get following operational properties, for every , and .
In [17, 18], the definition of a fuzzifying matroid on was given.
Definition 2. (cf. [17, 18]) A fuzzifying matroid is a finite set and a mapping from to (called fuzzy family of independent sets) such that (FI1)–(FI3) are satisfied, written as . For any two subsets of , (FI1) ; (FI2) if ; (FI3) (A {e}) ≥ if .For a fuzzifying matroid, the following characterization theorem holds.
Theorem 1 (cf. [17]). If is a mapping from to , then, is a fuzzifying matroid, which is equivalent to each of the following statements.(1){} is a family of matroids;(2){} is a family of matroids.It is also important to notice that a fuzzifying matroid can be defined equivalently as a pair , where is a mapping from to satisfying two axioms [19]: (FC1) for any , satisfies circuits axioms (C1)–(C3); (FC2) for any subset of , implies . In fact, let be a fuzziying matroid (i.e., satisfies (FI1)–(FI3)), the mapping defined by for any subset of such that {}, not only is the circuit-map of (i.e., satisfies (FC1)-(FC2)) but also for any . Conversely, let be a fuzziying matroid (i.e., satisfies (FC1)-(FC2)), the mapping defined by {} satisfies (FI1)–(FI3)) and .
The fuzzifying rank function was also defined.
Definition 3. (cf. [17, 23]). A mapping from to is called fuzzifying rank function of a given fuzzifying matroid , if for any subset of and any natural number such that {}.
Theorem 2 (cf. [17, 23]). The fuzzifying rank function for a fuzzifying matroid has the properties:(1).(2).Furthermore, for any fuzziying matroid (i.e., satisfies (FI1)–(FI3)), its fuzzifying rank function possesses some features, shown as, for any two subsets of , (FR1) ; (FR2) implies ; (FR3) . Conversely, for a fuzzifying rank function (i.e., satisfies (FR1)–(FR3)), the mapping defined by satisfies (FI1)–(FI3)) and .
Definition 4. (cf. [25]) For any fuzzifying matroid , a new mapping from to is given by for any subset of such that . It is proved that satisfies (FR1)–(FR3)), and thus, satisfies (FI1)–(FI3)), denoted by . Hence is a fuzzifying matroid, which is called the dual of , written as . In addition, the dual of is denoted by .
Theorem 3. (cf. [25]) Suppose that is a fuzzifying matroid, then, is equivalent to each of the following statements:(1) for any element in ,(2) for any element in .
3. The Course of Discovering the Notion of Connected Degree of Fuzzifying Matroids
We say that a graph is 2-connected if any two distinct edges are contained in a cycle. This notion (actually many others) is easy-to-understand to many students and practitioners. A natural question is that can they discover, starting from such a notion and its relevant results, some innovative things which are interesting, meaningful, or needed for themselves? This paper will exemplify a positive answer. Precisely, it will generalize (by using plausible reasoning method [26]) the notion of 2-connectedness of graphs to a very extended and useful notion (called connected degree of fuzzifying matroids) and present some properties of connected degree of fuzzifying matroids.
We will sketch the course of discovering the notion of the connected degree of fuzzifying matroids which may be of reference value to both our research successors and out-of-the-box researchers.
Step 1. Find out a characterization 2-connectedness of graphs (from all) start from which the notion described can be generalized: for a graph, written , we call that it is 2-connected if its any two distinct edges are contained in a cycle [2, 3], i.e., for any two different elements in , one element in (the set of all cycles of ) can be found to include and , simultaneously.
Step 2. Since matroid concept can be regarded as a generalization of graph concept (for a graph , the collection of its cycles is represented as , which is exactly the family of circuits for the cycle matroid of , thus, can also denoted equivalently as . So, we can think of as a matroid ), it is natural to generalize the notion of 2-connectedness from the class of graphs (written as ) to the class of matroids (written as ).
Step 3. It is also reasonable saying the special matroid is connected when is 2-connected, i.e., for any two different elements in , one element in can be found to include and simultaneously (see Step 1). Generalization of this can be taken as a definition of connectedness of a matroid (see Step 4).
Step 4. We call that a matroid is connected if, for any two different elements in , one element in can be found to include and simultaneously [2, 3].
Step 5. As a matroid is a special fuzzifying matroid (for any matroid or , we do not distinguish between and and between and , where and are characteristic functions of and , respectively), we naturally extend the notion of connectedness from the class of matroids to that of fuzzifying matroids (written as ).
Step 6. It is reasonable saying the special fuzzifying matroid is connected when the corresponding matroid is connected. In other words, for any {} E, {} = 1. Generalization of this can be taken as a definition of connectedness of a fuzzifying matroid (see Step 7).
Step 7. A fuzzifying matroid is called connected (or 2-connected) if for any {} E, {} = 1 (see the following Theorem 4 for a characterization).
Step 8. Find out a characterization (i.e., the separator characterization) of the connectedness of matroids (from all) start from which the notion described can be generalized: A matroid is disconnected if and only if (briefly, iff) a proper subset of can be found to be a separator of . Generalization of this can be taken as a definition of connected degree of a fuzzifying matroid (see Step 12).
Step 9. Recall and redescribe the definition of the separator (so that connectedness of a matroid can be generalized to connected degree of a fuzzifying matroid). Suppose that is a matroid, a subset of is called a separator of (in other words, the degree to which is a separator of is equal to 1, denoted by ) iff for any there must be or , that is, whenever is a subset of not included in both and (i.e., { is a subset of not included in both and } = 1). Compare the values of and { is a subset of not included in both and }, we get = { is a subset of not included in both and }. It is natural to generalize the equation from a matroid to a fuzzifying matroid (see Step 10).
Step 10. Given a fuzzifying matroid and a subset of , we call that = { is a subset of not included in both and } is the degree to which is a separator of .
Step 11. A matroid is disconnected (that is to say the connected degree of is equal to 0, denoted by ) if a proper subset of can be found to be a separator of (i.e., { {}} = 0). Compare the values of and { {}}, we get { {}}. It is natural to generalize the equality from a matroid to a fuzzifying matroid (see Step 12).
Step 12. Given a fuzzifying matroid , we call { {}} is the connected degree of .
4. Characterizations and Rationality of Connectedness of Fuzzifying Matroids
Although a fuzzifying matroid can be defined as both a pair and a pair , we find that the circuit-map axioms (FC1)-(FC2) of a fuzzifying matroid is not a natural way to generalize the circuits axioms (C1)–(C3) of a matroid. Hence, in the subsequent discussion, we always suppose that a fuzzifying matroid and a matroid are pairs and , respectively.
The following Definition 5, Theorem 4, and Corollary 1 (which characterize the connectedness of fuzzifying matroids, see Step 7 for the definition) show that the connectedness of fuzzifying matroids is in harmony with that of matroids.
Definition 5. Amapping from to is the circuit-map of a given fuzzifying matroid which is said to be connected if for any two elements in such that {} = 1.
Theorem 4. A fuzzifying matroid is connected, which is equivalent to is connected.
Proof. If is connected, then, for any two elements in , a subset of can be found to satisfy and include and , and thus by the property of . This implies that is connected. Conversely, if is connected, then, for any two elements in , one element in can be found to include and , and thus {} = 1 − 0 = 1. Hence, {} = 1, which implies that is connected.
Notice that, the fuzzifying matroid induced by a fuzzy family of circuits is introduced [27], it has the following properties, for any two elements in , implies . Thus, Theorem 4 deduces the following corollary.
Corollary 1. A fuzzifying matroid induced by a fuzzy family of circuits is connected if and only if is connected .
A fuzzy graph [28] is a set of nodes together with two fuzzy sets and on and , respectively, satisfied for any pair of elements of , , written as . Graph is called the underline graph of , and we identify that and are characteristic functions of and , respectively. When and take no value in , equals and so it is a graph, which implies that the class of fuzzy graphs contains that of graphs. Furthermore, the concept of connectedness can be extended from graphs to more general a fuzzy graphs. In one way, a fuzzy graph is 2-connected, which is defined as its underline graph is 2-connected. Actually, each fuzzy graph can induce a fuzzifying matroid which is the set of edges in and a mapping from to defined as for every subset of such that { included in A}, called fuzzifying cycle matroid of . Therefore, the notion of 2-connectedness for fuzzy graphs can be generalized to that of connectedness for their fuzzifying cycle matroids, and then to that of connectedness for more general fuzzifying matroids (see the following Theorem 5) which means the connectedness of fuzzifying matroids is also in harmony with 2-connectedness of fuzzy graphs.
Lemma 1. and are the fuzzifying cycle matroid and the cycle matriod, respectively, induced by a fuzzy graph and its underline graph , we have .
Proof. , i.e., , thus, a number in can be found such that . Suppose that contains a cycle of , then contains a cycle of since . There is a contradiction. Therefore, contains and any cycle of , i.e., .
, i.e., contains and any cycle of is not included in . We can find a number in (0,1) satisfying because the range of is a finite set. Hence, also contains and any cycle of is not also included in . This implies that , we have .
By Theorem 4 and Lemma 1, we get the fuzzifying cycle matroid is connected iff is connected iff is connected iff is 2-connected iff is 2-connected. Hence, the following theorem holds.
Theorem 5. Fuzzifying cycle matroid induced by a fuzzy graph is connected, which is equivalent to is 2-connected.
5. Connected Degree of Fuzzifying Matroids
Recall that the connected degree of a fuzzifying matroid is defined by using the notion of the degree to which a set is a separator (see Steps 10 and 12). In this section, we study the connections between the notions of the degree to which a subset is a separator, connected degree, and connectedness of a fuzzifying matroid. We also obtain a generalization of a classical result on separators of matroids (see Theorem 7).
Definition 6. Define the degree to which is a separator of a given fuzzifying matroid as { is a subset of not included in both and }.
Theorem 6. The following statements hold, see Definition 6 for this symbol .(1)If , then, is a separator of .(2)If , then, is not a separator of .(3)If , uncertain.(4) if and only if is not a separator of .
Proof. (1)Suppose that is not a separator of , then, one element can be found to be not included in both and ; hence, and then by the definitions of and .(2)Suppose that , i.e., { is a subset of not included in both and } = , then, for some subset of not included in both and , which implies this subset . In other words, for some subset of not included in both and , i.e., .(3)Take a counter example. A finite set = {} together with a mapping from to defined as ({}) = 1, ({e2}) = ({e3}) = , ({}) = ({}) = ({}) = ({}) = , form a fuzzifying matroid. Then, (if ), {{}, {}, {}} (if ), {{e2}, {e3}} (if ) by the definitions of and , and thus by the definition of , (if {, {}, {}}), (if {{e2}, {e3}}), (if {{}, {}, {}}). Let A = {e1}, by Definition 6, { = {}, {}, {}} = . We can check that is not a separator of although it is a separator of .(4)If , then, is not a separator of by (2). Conversely, if is not a separator of , then, by (1), i.e., .A classical result on separators of matroids: for any subset of , it is a separator of a given matroid , which is equivalent to the rank function satisfies . Its promotion is as follows:
Theorem 7. The following conditions are equivalent, among them, symbols and refer to Definition 6 and Definition 3, respectively.(1);(2) is a separator of for each ;(3);(4) is a separator of for each .
Proof. . If , then, for each , , thus, is a separator of by Theorem 6 (1). Conversely, if (2) holds, i.e., is a separator of for each . Assume that , then, is not a separator of by Theorem 6 (2), which contradicts (2). Hence, .
By Theorem 2 and the property of the addition of fuzzy natural numbers, we have
. is a separator of for each for each for each for each .
. for each for each for each is a separator of for each .
Definition 7. { {}} is called connected degree of a given fuzzifying matroid , where is defined in Definition 6.
The connection between the connected degree of a fuzzifying matroid and the degree to which a subset is a separator is described in the following theorem.
Theorem 8. The following statements are true, where symbols and are defined in Definition 6 and Definition 7, respectively.(1)If , then for each nonempty proper subset of .(2)If , then for some nonempty proper subset of .(3)If , uncertain.(4) if and only if for some nonempty proper subset of .(5) if and only if for each nonempty proper subset of .
Proof. (1)Suppose that there exists a nonempty proper subset of such that , then, { {}} ≤ a.(2)Suppose that , i.e., { {}} = a, then for some {}, that is, for some nonempty proper subset of .(3)Consider the fuzzifying matroid in [27], it is a set E = {} and a mapping from to defined as {}. Here, is constant in (resp., , ), , , and are matroids with the families of circuits {{}}, {{}, {}, {}}, and {{}, {}, {}, {}, {}, {}}, respectively. Then, by the definition of , (if {}), (if {{}, {}}), (if {{}, {}, {}, {}, {}}), and 0 (otherwise). Thus, by Definition 6, (if {}), (if {{}, {}}), (if {{}, {}, {}, {}}), and 0 (otherwise). Therefore, by Definition 7, . For , Sep ({e2}) = although for and every nonempty proper subset of .(4)If , then, there exists a nonempty proper subset of such that by (2). Conversely, if for some nonempty proper subset of , then, by (1), i.e., .(5) { {}} = 1 ( {}) , ( {}).The following results show that the relationship between connected degree of a fuzzifying matroid and connectedness of its corresponding level matroids.
Theorem 9. The following statements are also true, among them, symbol refers to Definition 7, which is the connected degree of a fuzzifying matroid .(1)If , then, is not connected for any . However, the converse is not true.(2)If if and only if is connected.
Proof. (1)Suppose that , by Definition 7, there exists {} such that , and thus ; hence, by Theorem 6. This implies that is not a connected matroid. Consider the fuzzifying matroid defined in the proof of Theorem 8 (3). Obviously, three level matroids are all not connected. Take an − {1/2}; then, is not connected for any , but .(2)It follows from Theorems 6 (4) and 8 (5).An immediate consequence of Theorems 5 and 9 is the following Theorem 10 which shows the connections between the connected degree of a fuzzifying matroid and connectedness of a fuzzifying matroid.
Theorem 10. A fuzzifying matroid is connected, which is equivalent to its connected degree .
The rest of this section discusses the fuzzy analogs of the results: A separator in a matroid is also a separator in its dual, and the connectedness of a matroid is equivalent to that of its dual.
Theorem 11. The symbols and represent a fuzzifying matroid and its dual , if , then, the following statements hold.(1) for each , where and are the degrees to which is a separator of and .(2).(3) is connected, which is equivalent to is connected.
Proof. (1)Suppose that , then, is not a separator in by Theorem 6 (2), so it is not in the dual matroid , nor in by and Theorem 3. By Theorem 6 (1), , i.e., . Conversely, since .(2)It is an immediate consequence of (1) by the definition of connected degree for fuzzifying matroids.(3)It is obvious by Theorem 10 and (2).
Remark 1. The characterization of the disconnectedness for a given fuzzifying matroid induced by a fuzzy family of circuits in [27] is as follows: is disconnected if and only if there is a nonempty proper subset of such that . From Theorems 7 and 8, it can also be characterized by the connected degree in this paper, is disconnected if and only if , that is, is connected if and only if . From this, we can see the connected degree introduced in this paper unifies the connectedness in [27].
We now describe the concepts of this paper by various examples.
Example 1. A simple class of fuzzifying matroids are fuzzifying uniform matroids. It is a fuzzifying matroid satisfying for any , implies [29].
Consider the example that a fuzzifying uniform matroid is a finite set E = {} and a mapping from to defined as , ({e1}) = II ({e2}) = II ({e3}) = 0.5, ({}) = II ({}) = II ({}) = 0.3, ({}) = 0.2. So, by the definition of circuit-map, (if {}), 0.5 (if {{e1}, {e2}, {e3}}), 0.7 (if {{}, {}, {}}), and 0.8 (if {{}}). By Definition 6, we have Sep ({e1}) = { = {}, {}, {}} = . Similarly, we also have if {{}, {}, {}, {}, {}}. Hence, { {}} = 0.8.
Example 2. Another simple and more general class of fuzzifying matroids is fuzzifying paving matroids. It is a fuzzifying matroid satisfying for any , implies [29].
Consider the fuzzifying matroid given in the proof of Theorem 6 (3), which is a fuzzifying paving matroid rather than a fuzzifying uniform matroid. Similarly, we also have if = {e2}, {e3}, {}, {} or {}. Hence, { {}} = 2/3.
Example 3. Similarly, we can use the examples of fuzzifying acyclic matroids and fuzzifying simple matroids introduced in [29] to illustrate the concepts of this paper, which are omitted here.
Example 4. The very important class of fuzzifying matroids are derived from fuzzy graphs, which have been discussed in Section 4 in detail. In addition, we can also proved that (if does not contain any cycle), 0 (otherwise). Note that when for any , a fuzzy graph is abbreviated to .
Consider the example that a fuzzifying cycle matroid induced by in Figure 1, and record it as , where {} is the edges set in and , , , . Then, (if {{}, {}}), 0.1 (if {{e1}, {}, {}, {}, {}, {}}), 0.2 (if {{e2}, {}, {}, {}}), 0.3 (if {{e3}, {}}), 0.4 (if {{e4}}), and 1 (if {}). By the definition of circuit-map, (if {{}}), 0.9 (if {{e1}}), 0.8 (if {{e2}}), 0.7 (if {{e3}}), 0.6 (if {{e4}}), and 0 (otherwise). By Definition 6, Sep ({}) = { = {}, {}, {}, {}, {}, {}, {}} = 1. Hence, { {}} = 0.
6. Conclusions
This paper exemplifies an application of Polya’s plausible reasoning method (mainly analogous and generalization) in matriod theory, where we obtain some new results on 2-connectedness of fuzzifying matroids but just start from the 2-connectedness of graphs (a notion popularized for undergraduate students). We show some properties of connectedness and connected degree of fuzzifying matroids and the reasonability of the definition of connected degree and enrich the theory of fuzzifying matroids. The course of discovering the notion of connected degree of fuzzifying matroids and related results is given clearly and relatively understandably which may be benefiticial to mathematics enthusiasts and researchers in other areas who are interested to discover. There are also some work to be performed further. For example, (1) For the notion of connectedness, a fuzzy graph is either connected or not, we can consider the fuzziness degree of connectedness for a fuzzy graph analogously. (2) We can define and study connected degree of Goetschel–Voxman fuzzy matroids analogously; we can even define and study connected degree of -matroids introduced in [18, 30] (a generalization of Goetschel–Voxman fuzzy matroids) analogously. (3) As we know, as an application of fuzzifying matroids, the fuzzifying greedy algorithm was presented in [23, 24], fuzzifying matroids are precisely the structures for which the very simple and efficient fuzzifying greedy algorithm works. Whether connectedness of fuzzifying matroids could be applied to some fuzzy combinatorial problems, we shall consider this problem in the future. Of course, applications in decision-making problems as in [31–34] should also be considered; (4) We are also extending our work to -polar fuzzifying matroids by imitating reference [15].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work was supported by the National Natural Science Foundation of China (grant no. 11771263, 61967001, and 61807023) and the Fundamental Research Funds for the Central Universities (grant no. GK202105007 and GK201702011).