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Covering is a widely used form of data structures. Covering-based rough set theory provides a systematic approach to this data. In this paper, graphs are connected with covering-based rough sets. Specifically, we convert some important concepts in graph theory including vertex covers, independent sets, edge covers, and matchings to ones in covering-based rough sets. At the same time, corresponding problems in graphs are also transformed into ones in covering-based rough sets. For example, finding a minimal edge cover of a graph is translated into finding a minimal general reduct of a covering. The main contributions of this paper are threefold. First, any graph is converted to a covering. Two graphs induce the same covering if and only if they are isomorphic. Second, some new concepts are defined in covering-based rough sets to correspond with ones in graph theory. The upper approximation number is essential to describe these concepts. Finally, from a new viewpoint of covering-based rough sets, the general reduct is defined, and its equivalent characterization for the edge cover is presented. These results show the potential for the connection between covering-based rough sets and graphs.

Covering is an extensively used form of data representation. As a generalization of classical rough set theory [

Graphs are important discrete structures consisting of vertices and edges that connect these vertices, and they can well describe the relationship among objects. Problems in almost every discipline can be addressed using graph models. However, some important problems in graphs are NP-hard optimization ones such as finding the minimal vertex cover [

In this paper, some graph concepts including vertex covers, independent sets, edge covers, and matchings are equivalently formulated using covering-based rough sets. First, a graph is represented with a covering, and an isomorphism from simple graphs without isolated vertices to a special type of coverings is constructed. Second, vertex covers, edge covers and matchings are equivalently described through the upper approximation number and independent sets with the lower approximation. Third, edge covers are also characterized by general reducts which are generalizations of the covering reduct. Furthermore, some graph problems are transformed into ones in covering-based rough sets. For instance, finding a minimal edge cover of a graph is equivalently converted to finding a minimal reduct of a covering.

There are numerous applications concerning connections between covering-based rough sets and graphs. For example, chemical classification, job assignment, and production process arrangement can be well modeled by graphs. However, many of these practical problems are NP-hard. Due to equivalent characterizations of covering-based rough sets and graphs, these problems can be converted and addressed under the framework of covering-based rough sets. In this way, heuristic reduction algorithms [

The rest of this paper is arranged as follows. Section

This section introduces some fundamental definitions related to covering-based rough sets and graphs.

As a generalization of partitions, coverings are with strong applicability and high universality. For example, a course consists of a number of students, and all courses form a covering of all students. Since a student can choose several courses, the covering is not necessarily a partition of all students. All students compose the set of research objects, namely, universe of discourse.

Let

Each subset in a covering is called a covering block also called a basic concept in knowledge discovery, and each subset of a universe is called a concept. An important idea of covering-based rough sets is to approximate a concept using some basic ones. This idea is implemented by a pair of lower and upper approximations.

Let

The lower approximation provides a certainty, and the upper one offers a probability. If we suppose that the universe is all students selecting courses and the covering is the set of all courses, then for any subset of students, its lower approximation is those courses which all students select in the subset, and its upper approximation is those courses which some students select in the subset.

Some covering blocks may be redundant; in other words, removing them has little effect on the approximation accuracy. For example, if a course is a required one and all students must select it, then removing it does not affect the lower approximation. The concept of the reducible element was proposed to describe those redundant covering blocks.

Let

The reducible element well reveals the relationship between coverings and their lower approximations. In fact, two coverings generate the same lower approximation if and only if their reducts are the same.

Graphs are discrete structures to model the correlation between data. Theoretically, a graph is an ordered pair consisting of vertices and edges that connect these vertices.

A graph

The simple graph is a main research objective of graph theory, and many practical problems can be represented with a simple graph.

A simple graph is a graph without loops or multiple edges, where a loop is an edge whose endpoints are equal and multiple edges are edges having the same pair of endpoints.

Graphs are visual and efficient tools to reveal the interrelation between data. Different graphs may have a certain internal relation. For this reason, isomorphism is introduced to express the relationship between graphs.

An isomorphism from one simple graph

In a graph that represents a road network (with straight roads and no isolated vertices), we can interpret the problem of finding the minimal vertex cover as the problem of placing the minimal number of policemen to guard the entire road network. It is noted that an isolated vertex of a graph is a vertex that is not an endpoint of any edge.

Let

Let

Let

Let

Covering-based rough sets are studied qualitatively, and they are short of quantitative approaches. The following definition presents a measure to conduct the quantitative analysis. This measure is also a bridge between covering-based rough sets and graphs.

Let

The above definition presents the upper approximation number with respect to a covering. Note that this concept can be defined similarly on any subcovering or family of subsets of a universe.

Let

The following proposition shows some properties of the upper approximation number and its connections with covering-based rough sets.

Let

if

for all

When a covering is included in another one, their corresponding upper approximation numbers present a similar characteristic.

Let

A partition can be equivalently characterized by the upper approximation number. A covering is a partition if and only if the upper approximation number of the set with only one element is equal to one.

Let

The upper approximation number plays an important role in both conducting quantitative analyses on covering-based rough sets and building connections between them and graphs.

In this section, we convert a graph to a covering in order to construct a platform for solving graph problems using covering-based rough sets. The following definition points out an approach to representing a simple graph with a family of subsets of vertices.

Let

According to Definition

Let

(

(

According to Proposition

In the rest of this paper, a graph is a simple one without isolated vertices unless otherwise stated. The following example illustrates the graph and its covering.

Let

A simple graph.

The following proposition explores the relationship between two graphs inducing the same covering. In fact, two different graphs induce the same covering if and only if they are isomorphic. Isomorphic graphs can be regarded as the same in graph theory. In other words, in an isomorphic sense, a one-to-one correspondence between coverings whose elements have only two objects and graphs is established.

Let

Independent sets, vertex covers, matchings, and edge covers of a graph are important concepts, and they stem abstractly from some practical problems. For instance, the maximal matching is from the job assignment problem. But some of them such as finding a minimal vertex cover are typical examples of NP-hard optimization problems. Therefore, there is much necessity to equivalently characterize these concepts. The following proposition presents a sufficient and necessary condition for the vertex cover through the upper approximation number.

Let

(

(

In fact, in Proposition

Let

According to Definition

The above proposition shows that finding a minimal vertex cover of a graph is equivalently transformed into finding a minimal subset whose upper approximation number is equal to the cardinality.

Let

Maximal vertex cover.

The following proposition indicates that an independent set of a graph can also be formulated equivalently with the lower approximation operator. In fact, a subset of vertices of a graph is an independent set if and only if its lower approximation is empty.

Let

(

(

Let

According to Definition

The above proposition indicates that finding a maximal independent set is converted to finding a maximal subset keeping its lower approximation empty. The following example illustrates independent sets of a graph and its connection with the lower approximation.

Let

Maximal independent set.

A matching of a graph can also be represented with the upper approximation number. In fact, an edge subset of the edges of a graph is a matching if and only if the upper approximation number of these sets having only one element is not less than one.

Let

Furthermore, the perfect matching of a graph is concisely characterized by the upper approximation number.

Let

Let

Let

A maximal matching.

Let

(

(

Let

This section presents another view to represent graph problems with covering-based rough sets. First of all, we define the generally reducible element of a covering, which is an extension of the reducible element in the literature [

Pawlak defined category reducts for knowledge reduction and rule extraction [

Let

Let

Let

The following proposition shows that the generally reducible element is an extension of the reducible element. In other words, if a covering block is reducible, then it is generally reducible.

Let

The generally reducible element is an extension of the reducible element; however, the general reduct is not an extension of the reduct. The following counterexample illustrates this argument.

Let

The following proposition explores the relationship between general reducts and edge covers of a graph. In fact, an edge subset of a graph is an edge cover if and only if it contains at least one general reduct.

Let

(

(

According to Proposition

Let

According to Proposition

Matchings of a graph can also be described using the generally reducible element.

Let

Proposition

Let

In this paper, we presented equivalent characterizations for some important problems in graph theory from the viewpoint of covering-based rough sets. These problems included vertex covers, independent sets, edge covers, and matchings of a graph, where finding a minimal vertex cover was NP-hard. The equivalent characterizations indicated covering-based rough sets approaches to these problems. Moreover, graph concepts such as vertex connectivity and graph approaches such as shortest path algorithms were available to study covering-based rough sets. In future works, we will apply these interesting theoretical results not only to algorithm design for some graph problems but also to unsupervised learning, especially bipartite graph clustering and spectral clustering.

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 Grants nos. 2011J01374 and 2012J01294, and the Science and Technology Key Project of Fujian Province, China, under Grant no. 2012H0043.