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• Outline Introduction Preliminaries Main Results Conclusion References A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC NUMBER OF A FUZZY GRAPH Isnaini Rosyida Widodo, Ch.Rini Indrati, Kiki Ariyanti Sugeng IICMA 2013 UGM, Yogyakarta Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References 1 Introduction 2 Preliminaries 3 Main Results A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set 4 Conclusion 5 References Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Introduction Abstract A δ-chromatic number of a fuzzy graph (δ ∈ [0, 1]) has been introduced by Cioban [1]. However, he did not deﬁne a fuzzy chromatic set of a fuzzy graph yet. In this paper we have a new approach in constructing a fuzzy chromatic set of a fuzzy graph. The fuzzy chromatic set is constructed through the δ-chromatic number. Further, we show that the fuzzy chromatic set satisﬁes properties of a discrete fuzzy number and then it is called fuzzy chromatic number. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Introduction Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Introduction Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Preliminaries Fuzzy Set (Zadeh, 1965) ˜ Let X be a space of objects. A fuzzy set A in X is a set of the form {(x, µA (x)) : x ∈ X }, ˜ ˜ where µA : X → [0, 1] is a membership function of the fuzzy set A. ˜ Fuzzy Set (Zadeh, 1965) ˜ ˜ The support of fuzzy set A, denoted by S(A), is the crisp set given ˜ by S(A) = {x ∈ X |µA (x) > 0}. Let α ∈ (0, 1], the α-cut of the ˜ ˜ ˜ fuzzy set A is the crisp set Aα = {x ∈ X |µA (x) ≥ α}. While ˜ ˜ ˜ A0 = {x ∈ X |µA (x) > 0} is the closure of support A. ˜ Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Preliminaries Fuzzy number ˜ A fuzzy set A, deﬁned on the set of real numbers R, is said to be a fuzzy number if it satisﬁes the following conditions (Bector and Chandra, 2005): ˜ i A is normal, i.e.∃x0 ∈ R such that µ ˜ (x0 ) = 1. A ˜ ii Aα is a closed interval for every α ∈ (0, 1]. ˜ iii The support of A is bounded . Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Preliminaries Discrete fuzzy number (Wang, 2008) ˜ Let C ⊆ R be a countable set. A fuzzy set A is called discrete fuzzy number in C if it satisﬁes the following conditions [2]: ˜ i the set A0 ⊂ C and it is ﬁnite; ii there exists x0 ∈ C such that µA (x0 ) = 1; ˜ iii µA (xs ) ≤ µA (xt ) for any xs , xt ∈ C with xs ≤ xt ≤ x0 ; ˜ ˜ iv µA (xs ) ≥ µA (xt ) for any xs , xt ∈ C with xs ≥ xt ≥ x0 . ˜ ˜ Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Preliminaries Fuzzy Graph ˜ ˜ Kaufmann (1973): A fuzzy graph G (V , E ) is a graph having a ˜ with a membership crisp vertex set V and a fuzzy edge set E function µ : V × V → [0, 1] (Sunitha, 2001). A graph G (V , E ) will be called a crisp graph. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Preliminaries Fuzzy independent vertex set (Bershtein and Bozhenuk,1999) ˜ ˜ ˜ Given a fuzzy graph G (V , E ). Let G (S, ES ) be a fuzzy subgraph ˜ (V , E ). A set S ⊆ V will be called fuzzy independent vertex ˜ of G set (fuzzy internal stable set) with the degree of independence α(S) = 1 − max{µ(x, y )|x, y ∈ S}. ˜ The separation degree of a fuzzy graph G with k colors is deﬁned as L = min{α(Vi )|i = 1, ..., k}. Maximal fuzzy independent vertex set ˜ ˜ Let G (V , E ) be a fuzzy graph with n vertices, A subset S ⊆ V is called a maximal fuzzy independent vertex set, with the degree α(S ), if the condition α(S ) < α(S ) is true for any subset S ⊂S . Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Preliminaries Fuzzy chromatic set (Bershtein and Bozhenuk,2001) A fuzzy set γ = (k, Lγ (k))|k = 1, ..., n , where Lγ (k) denotes the ˜ ˜ ˜ ˜ separation degree of G with k colors, is called a fuzzy chromatic ˜ set of G if and only if there is not more than k -maximal fuzzy independent vertex sets X1 , X2 , . . . , Xk (k ≤ k) with the degrees of independence respectively α1 , α2 , . . . , αk , that satisfy: 1) min{α1 , α2 , . . . , αk } = Lγ (k) ˜ 2) ∪j=1,...,k Xj = V ; 3) there do not exist another family {X1 , X2 , . . . , Xk } with k ≤ k for which min{α1 , α2 , . . . , αk } > min{α1 , α2 , . . . , αk } and the condition 2) is true. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Preliminaries δ-Fuzzy independent vertex set Given δ ∈ [0, 1], a δ-fuzzy independent vertex set A is a set where µ(u, v ) ≤ δ for all u, v ∈ A. The δ-fuzzy independent vertex set A will be denoted as S δ . δ-Coloring ˜ ˜ The δ-coloring of a fuzzy graph G (V , E ) is a partition of V into k δ , ..., S δ } such that S δ ∩ S δ = ∅ for all i = j and subsets {S1 i j k δ δ S1 ∪ ... ∪ Sk = V . ˜ ˜ The δ-chromatic number of G , denoted by χδ (G ), is the smallest natural number k so that such partition is possible. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Preliminaries Example: δ-coloring . For δ = 0: χδ = 3; For δ = 0, 3: χδ = 2 Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set A new approach New construction for fuzzy chromatic Set ˜ ˜ Let G (V , E ) be a fuzzy graph with n vertices. Let δ ∈ [0, 1]. The ˜ fuzzy chromatic set of G , denoted by χ(G ), is a fuzzy set ˜ ˜ ˜ ) = {(k, L(k))} where χ(G ˜ ˜ L(k) = max{1 − δ|χδ (G ) = k, k = 1, . . . , n}. (1) The value L(k) represents the membership grade of the chromatic number k in the fuzzy chromatic set χ. ˜ Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set Example: Fuzzy chromatic set ˜ Figure : A fuzzy graph G with µ(V × V ) = {0, 0.2, 0.3, 0.5, 0.6, 1}. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set A new approach Fuzzy chromatic set: the new approach ˜ ˜ Considered a fuzzy graph G (V , E ) in Figure 1. For δ = 0, the partition of V is {{B, D}, {A}, {C }} and ˜ χδ (G ) = 3. For 0 ≤ δ < 0.3, the δ-chromatic number is 3. For δ = 0.3, the partition of V is {{B, C , D}, {A}} and ˜ χδ (G ) = 2. For 0.3 ≤ δ < 1, the δ-chromatic number is 2. ˜ For δ = 1, the partition is {A, B, C , D} = V and χδ (G ) = 1. ˜ The fuzzy chromatic set of G is χ(G ) = {(1, 0), (2, 0.7), (3, 1), (4, 1)} ˜ ˜ Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set Example: Bershtein’s method ˜ Table : Maximal fuzzy independent vertex sets of G D A B C x1 1 0 1 0 x2 0 1 0 0 x3 0 0 0 1 x4 0,8 0 0 0.8 x5 0,7 0 0,7 0,7 x6 0,5 0,5 0 0 x7 0,4 0,4 0,4 0 If k = 3 then the covering of all rows by 3 columns gives the set X1 , X2 , X3 with the separation degree L(3) = min{1, 1, 1} = 1. If k = 2, we have the set X2 , X5 with the separation degree L(2) = min{1, 0.7} = 0.7. ˜ The fuzzy chromatic set of G is χ(G ) = {(1, 0), (2, 0.7), (3, 1), (4, 1)} ˜ ˜ Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set Some properties‘of fuzzy chromatic Set Theorem 1 ˜ ˜ Given a fuzzy graph G (V , E ) with n vertices. Let δt ∈ [0, 1], there δt δt ˜ is a partition {S1 , . . . , Sk } which gives χδt (G ) = k and the greatest value of L(k) = 1 − δt if and only if there is k-maximal fuzzy independent vertex sets X1 , X2 , . . . , Xk with the degrees of independence respectively α1 , α2 , . . . , αk , that satisfy: 1) min{α1 , α2 , . . . , αk } = L(k). 2) ∪j=1,...,k Xj = V . 3) There do not exist another family {X1 , X2 , . . . , Xk } with k ≤ k for which min{α1 , α2 , . . . , αk } > min{α1 , α2 , . . . , αk } and the condition 2) is true. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set Outline proof of Theorem 1 (⇒). we consider two cases. for k = n. We have Siδt = {vi } and α(Siδt ) = 1 for all i = 1, . . . , n. While α(S δ ) < 1 for all Siδt ⊂ S δ . Thus, we have k-maximal fuzzy δ δ independent vertex sets: X1 = S1 t , . . . , Xk = Sk t . Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set Outline proof of Theorem 1 for k < n. There does not exist v ∈ V such that δ max{µ(x, y )|x, y ∈ S1 t ∪ {v }} = δt . Since otherwise, we will have δ δ δ a set S1 t = S1 t ∪ {v } with α(S1 t ) = 1 − δt and a partition δt δt δt (G ) = k and k < k. Thus, we have a {S1 , . . . , Sk } such that χ ˜ δ maximal fuzzy independent vertex set X1 = S1 t . If there is v ∈ V such that max{µ(v , x)|∀x ∈ Sjδt , j = 2, . . . , k} = δt , then we construct Xj = Sjδt ∪ {v } with α(Xj ) = 1 − δt . The process is continued until there does not exist w ∈ V such that max{µ(w , x)|∀x ∈ Sjδt } = δt . Thus, Xj , j = 2, . . . , k are maximal ˜ fuzzy independent vertex set in G . We can prove that the maximal fuzzy independent vertex sets δ δ X1 = S1 t , . . . , Xk = Sk t satisfy the properties 1)-3). Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set Outline proof of Theorem 1 (⇐). We will prove that there exists δt ∈ [0, 1] such that the δ δ ˜ partition {S1 t , . . . , Sk t } gives χδt (G ) = k and the greatest value of L(k) = 1 − δt . Let Xt ∈ {X1 , X2 , . . . , Xk } such that L(k) = min{α(X1 ), . . . , α(Xk )} = α(Xt ). By choosing δt = max{µ(u, v )|u, v ∈ Xt }, we have µ(u, v ) ≤ δt for all u, v ∈ Xi , i = 1, . . . , k and L(k) = α(Xt ) = 1 − max{µ(u, v )|u, v ∈ Xt } = 1 − δt . Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set Outline proof of Theorem 1 we can construct δt -fuzzy independent vertex sets: δ S1 t = X1 , δ S2 t = X2 − (X1 ∩ X2 ), δ S3 t = X3 − {(X1 ∩ X3 ) ∪ (X2 ∩ X3 )}, δ S4 t = X4 − {(X1 ∩ X4 ) ∪ (X2 ∩ X4 ) ∪ (X3 ∩ X4 )}, . . . δ Sk t = Xk − {(X1 ∩ Xk ) ∪ (X2 ∩ Xk ) ∪ . . . ∪ (Xk−1 ∩ Xk )}. δ δ δ Then the δt -fuzzy independent vertex sets S1 t , S2 t , . . . , Sk t satisfy: ∪j=1,...,k Sjδt = V and Siδt ∩ Sjδt = ∅ for all i = j. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set Some properties of fuzzy chromatic Set Theorem 2 ˜ ˜ Let G (V , E ) be a fuzzy graph. Given δ1 , δ2 ∈ [0, 1]. The value ˜ ˜ δ1 ≥ δ2 if and only if χδ1 (G ) ≤ χδ2 (G ). Theorem 3 ˜ ˜ Let G (V , E ) be a fuzzy graph. L(1) = 0 if and only if ∃u, v ∈ V such that µ(u, v ) = 1. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set Some properties of fuzzy chromatic Set Lemma 4 ˜ If δ = 0 then χ0 (G ) = χ(G ) = k and L(k) = 1. Lemma 5 ˜ Let i and j be δ-chromatic numbers of G . If i ≥ j then L(i) ≥ L(j). Lemma 6 ˜ If χδ (G ) = i has the degree L(i) = 1 and i = n then L(k) = 1 for all k ∈ {i + 1, ..., n}. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References A new Approach for fuzzy chromatic Set of a fuzzy Graph Some properties of fuzzy chromatic Set Some properties of fuzzy chromatic Set Theorem 7 ˜ ˜ Given a fuzzy graph G (V , E ) with n vertices. Let G (V , E ) be an ˜ underlying crisp graph of G . Let C be a set of the δ chromatic ˜ . The fuzzy chromatic set χ(G ) has the following number of G ˜ ˜ properties: i the support: Supp(χ) ⊂ C and it is ﬁnite; ˜ ii there exists k0 ∈ C such that L(k0 ) = 1; iii L(ks ) ≤ L(kt ) for any ks , kt ∈ C with ks ≤ kt < k0 ; iv L(ks ) = L(kt ) = 1 for any ks , kt ∈ C with ks ≥ kt ≥ k0 . Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References Conclusion In this paper, a new approach for ﬁnding a fuzzy chromatic set of a ˜ ˜ fuzzy graph G (V , E ) is introduced. The fuzzy chromatic set is constructed through the δ-chromatic number. We show that the new construction in this paper is equivalent with the fuzzy chromatic set resulted by Bershtein and Bozhenuk’s method ([1]). The proposed method is very easy to apply for solving the fuzzy chromatic set problems of a fuzzy graph. Further, we show that the fuzzy chromatic set of a fuzzy graph is a discrete fuzzy number and then it is called fuzzy chromatic number. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References References Bershtein, L. and Bozhenuk, A., ”A Colour Problem for Fuzzy Graph”, Proceedings of International Conference 7th Fuzzy Days, LNCS 2206, Dortmund. (2001), 500 - 505. Bershtein,L. and Bozhenuk, A, ”Maghout Method for Determination of Fuzzy Independent, Dominating Vertex Sets and Fuzzy Graph Kernels”, International Journal of General Systems. 30(1) (1999), 45-52. Bector, C.R. and Chandra, S., Fuzzy Mathematical Programming and Fuzzy Matrix Games, Springer Verlag, 2005. Biggs, N., Algebraic Graph Theory, Cambridge University Press, 1993. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References References Cioban, V., ”On Independent Sets of Vertices of Graph”, Studia Univ. Babes-Bolyai Informatica L.II. 1 (2007), 97 - 100. Cioban, V. and Prejmerean, V., ”Algorithms for Compute Independent Sets of Vertices in Graphs and Fuzzy Graphs”, Proceedings of IEEE International Conference on Automation,Quality and Testing,Robotics (ATQR), Cluj Napoca Romania. (2010), 1 - 5. Munoz, S., Ortuno, M.T., Javier, R., Yanez, J., ”Colouring Fuzzy Graph”, Omega: The Journal of Management Science. 33 (2005), 211 - 221. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC
• Outline Introduction Preliminaries Main Results Conclusion References References Sunitha, M.S., Studies on Fuzzy Graphs, Ph.D. Dissertation: Cochin University of Science and Technology, 2001. Wang, G., Zhang, Q., Cui, X., ”The Discrete Fuzzy Numbers on a Fixed Set With Finite Support Set”, Proceedings of IEEE Conference on Cybernetics and Intelligent Systems, Chengdu. (2008), 812 - 817. Zadeh, L.A, ”Fuzzy Sets”, Information and Control., 33 (1965), 338-353. Isnaini R, Widodo,Ch.Rini Indrati,Kiki A.Sugeng A NEW APPROACH IN CONSTRUCTING FUZZY CHROMATIC