Soft covering based rough graphs and corresponding decision making

Abstract Soft set theory and rough set theory are two new tools to discuss uncertainty. Graph theory is a nice way to depict certain information. Particularly soft graphs serve the purpose beautifully. In order to discuss uncertainty in soft graphs, some new types of graphs called soft covering based rough graphs are introduced. Several basic properties of these newly defined graphs are explored. Applications of soft covering based rough graphs in decision making can be very fruitful. In this regard an algorithm has been proposed.

In real situations, complexity and complications usually originate from uncertainty in the form of ambiguity. There are several real life problems involving uncertainty and vagueness where the classical mathematics is not successful and not absolutely prosperous. Most of our traditional and conventional mechanism of modeling, reasoning and computing are crisp, precise in character and deterministic. The dilemma and situations connecting with uncertainty are being handled by ancient and e ective tools of probability. The drawback of probability theory is that it is applicable only when the occurrence of events is strictly determined by chance. In conjunction with probability theory, many other theories like fuzzy set theory, intuitionistic fuzzy set theory, rough set theory, neutrosophic set theory, soft set theory and blend of some of these theories to handle uncertainty which arises due to vagueness, have been introduced (for further details see [4,5]). It is believed that an important epoch in the evolution of modern theory of uncertainty arising due to vagueness was the publication of the pioneering paper by Zadeh [7] in 1965. He has de ned fuzzy sets with an aspect to study, describe and develop mathematically those situations which are imprecise and de ned vaguely. Pawlak [6] introduced the concept of rough sets which is an excellent mathematical tool to handle with the given information and an access to ambiguity and equivocalness. The main signi cance of rough set theory is that it does not involve any additional information about the data, like membership in fuzzy sets. The rough sets theory is based on equivalence relations, which are now extended to the notion of covering based rough sets [8,9]. Probability theory, fuzzy set theory and rough set theory are di erent accessions to handle uncertainty, vagueness and imprecision. Each of these theories have their own restrictions and limitation. Many applications of these theories in data mining, pattern recognition, knowledge discovery and machine learning can be seen in [10][11][12][13][14][15][16][17]. While dealing with such theories, a question arises how to handle multi-attributes? Molodtsov [5] introduced the notion of soft sets to overcome the problem of dealing with attributes. This concept not only changed the role of above said theories as the sole representative of multiattributes but also recti ed in some disciplines to tackle many problems of uncertainty [18][19][20]. A number of applications, utilizations and practices have made with respect to multi-attributes modeling and decision making problems [24][25][26][27][28]. A useful and drastic theory has been established in [29][30][31] by connecting the covering soft sets to rough sets. Huge number of applications have been presented by many researchers in multi-attributes decision making problems, attributes reduction problems, data labeling problems, data mining problems and knowledge based systems [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. The concept of soft graphs and their di erent operations can be seen in [47]. These concepts were required to tackle multi-attributes problems related to the theory of graphs. A number of generalizations of soft graphs are available in the literature [48][49][50][51][52][53]. To strengthen and enhance the applicability of soft graphs, an innovative approach by combining rough set with soft graphs, called soft covering based rough graphs are introduced. In the present paper we initiate the study of new types of graphs called soft covering based rough graphs. Several properties of these graphs are explored. As an application of soft covering based rough graphs in decision making problems an algorithm is proposed.
The rest of the paper is organized as follows: In Section , some basic concepts are revised. Section is about basic de nitions and characterization of soft covering based rough graph, lower/upper S-soft vertex covering approximations, lower/upper Q-soft edge covering approximations, S-soft rough vertex covering graph, Q-soft rough edge covering graph, soft covering based rough graphs, and basic theory is discussed with examples. Section is devoted to present an application of soft covering based rough graphs in real life. To compute the e ectiveness of some diseases amongst colleagues working in same factory, an algorithm is developed in a realistic way, using simple digraph with vertices as colleagues and the edges as the interaction of these colleagues. Marginal fuzzy sets are de ned with the help of lower and upper soft rough approximations of the given graph and using marginal fuzzy sets as weights, persons at high risk of having given diseases are found. Conclusion of the paper is presented in Section .
(iv) covering soft vertex graph if λ (α) ≠ ∅ for all α ∈ T. In this case (λ,T) is called covering soft set over V, denoted by C V .
Denote by S = (V , C V ) and call it soft vertex covering approximation space.
De nition 7. Let S = (V , C V ) be a soft vertex covering approximation space and v ∈ V. Then the set for any X ⊆ V .
De nition 9. Let G S be an S-soft rough vertex covering graph. Then the S-roughness membership function of X ⊆ V is given by Thus if appr S (X) = appr S (X) then ξ G S (X) = , and the graph G S is soft vertex covering de nable, i.e., no roughness.  Table 1 v  Table such Since appr S (X) ≠ appr S (X) . So X is a soft vertex covering based rough set and G S : e , e , e , e , e , e , e }) and .
Thus there is no roughness.
De nition 10. Let G = G * , λ, µ,T be a full soft edge graph. Then G is called covering soft edge graph if µ (α) ≠ ∅ for all α ∈ T. In this case (µ,T) is called covering soft edge set over E, denoted by C E . Denote by Q = (E, C E ) and call it a soft edge covering approximation space.
De nition 11. Let Q = (E, C E ) be a soft edge covering approximation space and e ∈ E. Then the set is called soft minimal edge description of e ∈ E.
for the subset Y ⊆ E, are called the lower Q-soft edge covering and upper Q-soft edge covering approximations of Y, respectively. Also, are called Q-soft positive edge covering region, Q-soft negative edge covering region and Q-soft boundary edge covering region respectively. If appr and for any Y ⊆ E.
De nition 13. Let G Q be a Q-soft rough edge covering graph. Then the Q-roughness membership function of Y ⊆ E is given by Thus if appr Q (Y) = appr Q (Y) then ξ G Q (Y) = , and so the graph G Q is soft edge covering de nable, i.e., no roughness.

De nition 14.
A full soft graph G = G * , λ, µ,T is called covering soft graph if λ (α) ≠ ∅ and µ (α) ≠ ∅ for all α ∈ T.   Table 2 Let Y = {e , e , e , e } ⊆ E. Then As appr Q (Y) ≠ appr Q (Y) , so Y is a soft edge covering based rough set and G Q : .

De nition 15.
The G * -roughness membership function of any subgraph graph G ** = (X, Y) of G * is given by De nition 18. The lower and upper approximations of the soft covering based rough graph G = (G S , G Q ), is denoted and de ned as appr (G) = appr S (X) , appr Q (Y) and appr (G) = (appr S (X) , appr Q (Y)) respectively, for any X ⊆ V and Y ⊆ E.
Proposition 1. Let G = G * , λ, µ,T be a covering soft graph and S = (V , C V ) and Q = (E, C E ) be soft vertex covering approximation space and soft edge covering approximation space, respectively. Then appr(G ) and appr (G ) ⊆ appr(G ) where G and G are subgraphs of G * .

Proposition 2.
Let G = G * , λ, µ,T be a covering soft graph and let S = (V , C V ) and Q = (E, C E ) be soft vertex covering approximation space and soft edge covering approximation space, respectively. Then Proof. ( ) Suppose X is an S-soft vertex covering de nable. Then appr S (X) = appr S (X) and so appr S (X) = appr S (X) ⊆ X. Conversely suppose that appr S (X) ⊆ X for X ⊆ V . To prove X is an S-soft vertex covering de nable , we have to prove only appr S (X) ⊆ appr S (X) .
( ) Can be proved in similar way.

Proposition 6.
Let G = G * , λ, µ,T be a covering soft graph and Q = (E, C E ) be a soft edge covering approximation space. Then the following are equivalent: Proof. Similar to the proof of the Proposition 5.

Applications of Soft Covering Based Rough Graphs
One of the most important applications of rough sets is the decision making and after combining with soft sets, it has promoted to multicriteria group decision making. Many applications of multicriteria group decision making are available in literature which can be seen in [26,28,31,34,39]. In such applications, the decision making has not been involved by the interaction of the objects, while their individual performance/characteristics have been used. Initial evaluation results have been used to perform the algorithms of decision making, which are prescribed to a few number of elds. In this section, we use the soft covering based rough graphs to settle a real life medical diagnosis problem. The algorithm is described as follows: Let S = (V , C V ) be a soft vertex covering approximation space, then appr S (π (St)) = V P ∈ V : V P is an optimum candidate according to medical specialist S t , and appr S (π (St)) = V P ∈ V : V P is possibly an optimum candidate according to medical specialist S t .
Suppose Ω π(D) (V P ) and Ω π(D) (V P ) are two fuzzy sets for measure of optimality and possibly measure of optimality, respectively on E, of each object V P such that and where χ π(S t ) and χ π(S t ) are a kind of indicator functions, de ned by Clearly Ω π(D) (V P ) and Ω π(D) (V P ) represents the optimality and possible optimality of each object according to each medical specialist. Now consider the interaction of vertex Vp with vertex Vq and vice versa. The marginal weight function φ for each Vp can be computed by; for p = , , , ..., k, where For a threshold γ ∈ [ , ] , it can be seen that all persons V j are at optimum for all j, in which ψ (Vp) ≥ γ. The persons V k is the best optimal if ψ (V k ) = max p {ψ (Vp)} . This algorithm involved both the indivisual's evaluations as well as the e ects of interaction amongst the vertices/objects. This can be an interaction of two poles of transportation or network problems. One can apply this algorithm to other related problems. A real life application for diagnosing diseases from a group of people has been considered below. Suppose during the annual medical checkup, four viral diseases found in a group of people V = {V , V , V , ..., V }, through di erent sources such as insect bite, eating contaminated food, having sex with an infected person and breathing air polluted by a virus. The above process of infection results in a diversity of symptoms that vary in severity and character, depending upon the individual factor and the kind of viral infection. Suppose T = {d , d , d , d } is the set of parameters such that d represents " entering of virus in human body through insect bite ", d represents " entering of virus in human body through eating contaminated food ", d represents " entering of virus in human body through having sex with an infected person and d represents " entering of virus in human body through breathing air polluted by a virus. It is also assumed that a person V j may have more than one viral disease. Suppose G * is a simple digraph having vertex set V of persons, (λ, T) be covering soft set on V indicating which member has what disease and S = (V , C V ) be a soft vertex covering approximation space such that λ (d i ) : Let (µ, T) be a covering soft sets on E de ned by; Clearly G = G * , λ, µ,T is a soft graph, i.e., for each i = , , , ,  Let S = (V , C V ) be a soft vertex covering approximation space. Then appr S (π (S )) = {V , V , V , V , V , V , V , V , V , V , V , V }, appr S (π (S )) = ∅ = appr S (π (S )) and appr S (π (S i )) = V , for i = , , . Suppose Ω π(D) V j and Ω π(D) V j be two fuzzy sets for measure of optimality and possibly measure of optimality, respectively on E of each object V i such that

Conclusion
The applications of soft sets and rough sets that are available in the literature are usually based upon the individ's properties of the members of the universe with the given attributes. In decision making problems, the diversity of attribute/behavior and characteristics with the member's interaction have not been considered so far. In the present work, we introduced the notion of soft covering based rough graphs. We not only discussed the basic properties of such graphs but also formulated a prediction system to optimize the diagnosis process of diagnosing some diseases among the members working in a factory. This interaction may cause the spreadness of disease among the sta members. Using the concepts of lower/upper S-soft vertex covering approximations, the fuzzy sets Ω π(D) and Ω π(D) are introduced, while the marginal fuzzy sets φ r (Vp) and φ c (Vp) are used to nd the measure of interaction of any sta member Vp with Vq and vice versa. Finally the evaluation function has pointed out the optimal carriers of diseases. We hope our results will prove a foundation for decision making problems. In future work we will be working on decision making problems in which lower/upper covering soft edge approximations are used to optimize the algorithm and will try to use di erent techniques to replace the marginal fuzzy sets.