Interval-Valued Bifuzzy Graphs

Transcription

1 Interval-Valued Bifuzzy Graphs Muhammad Akram1 and Karamat H. Dar2 1. Punjab University College of Information Technology, University of the Punjab, Old Campus, Lahore-54000, Pakistan Department of Mathematics, G. C. University Lahore, P.O. Box 54000, Pakistan Abstract In this article, we introduce the notion of interval-valued bifuzzy graphs and describe various methods of their construction. We also present the concept of interval-valued bifuzzy regular graphs. Keywords: Graph, Interval-valued bifuzzy graphs, Interval-valued bifuzzy regular graphs Mathematics Subject Classification 2000: 05C99 1 Introduction In 1736, Euler first introduced the notion of graph theory. In the history of mathematics, the solution given by Euler of the well known Konigsberg bridge problem is considered to be the first theorem of graph theory. This has now become a subject generally regarded as a branch of combinatorics. The theory of graph is an extremely useful tool for solving combinatorial problems in different areas such as geometry, algebra, number theory, topology, operations research, optimization and computer science.

2 2 In 1975, Rosenfeld [8] introduced the concept of fuzzy graphs. The fuzzy relations between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts. Mordeson and Peng [4] introduced some operations on fuzzy graphs. Shannon and Atanassov [7] introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs, and investigated some of their properties. Recently, Akram and Dudek [1] have studied some properties of interval-valued fuzzy graphs. n this article, we introduce the notion of interval-valued bifuzzy graphs and describe various methods of their construction. We also present the concept of interval-valued bifuzzy regular graphs. 2 Preliminaries A fuzzy graph G = (J.L, v) is a non-empty set V together with a pair of functions J.L : V -+ [0, 1] and v: V X V -+ [0,1] such that v({x,y}):s min(j.l(x),j.l(y)) for all x, y E V. Fuzzy graph is a graph consists of pairs of vertex and edge that have degree of membership containing closed interval of real number [0,1] on each edge and vertex. In 1975, Zadeh [9] introduced the notion of interval-valued fuzzy sets as an extension of fuzzy sets [10] in which the values of the membership degrees are intervals of numbers instead of the numbers. An interval number D is an interval [a-, a+] with 0 :s a- :s a+ :s 1. The interval [a, a] is identified with the number a E [0,1]. D[O,l] denotes the set of all interval numbers. For interval numbers DJ = [ai ' btl and D2 = [az, btl, we define rmin(dj' D2 ) = rmin([al, btl [az, bi]) = [min{ ai' a2}, min{bi, btl], rmax(dj, D2 ) = rmax([al, btl [az, btl) = [max{al, az }, max{bi, bt}], DJ + D2 = [al + az - al. az, bi + bt - bi. btl, The interval-valued fuzzy set A in V is defined by A = {(x, [J.LA(X),J.L~(x)]): x E V}, where J.L:4(x) and J.L~(x) are fuzzy subsets of V such that J.L:4(x) :s J.L~(x) for all x E V. Definition 2.1. By an interval-valued fuzzy graph of a graph G* = (V, E) we mean a pair G = (A, B), where A = [J.L:4, J.L~] is an interval-valued fuzzy set on V and B = [J.L8, J.Ltl is an interval-valued fuzzy relation on E.

3 3 The notion of interval-valued intuitionistic fuzzy sets was introduced by Atanassov and Gargov in 1989 as a generalization of both interval-valued fuzzy sets and intuitionistic fuzzy sets. Gerstenkorn and Manko [6] re-named the intuitionistic fuzzy sets as bifuzzy sets in Definition 2.2. For a nonempty set G, we call a mapping A = (JiA, VA) : G -+ D[O, 1] x D[O, 1] an interval-valued bifuzzy set in G if Ji~(x) + v,:t(x) :-:; 1 and Ji;:;(x) + v;:;(x) :-:; 1 for all x E G, where the mappings JiA(X) = [Ji;:;(x),Ji~(x)] : G -+ D[O, 1] and VA(X) = [V;:; (x), v,:t(x)] : G -+ D[O, 1] are the degree of membership functions and the degree of nonmembership functions, respectively. We use 0 to denote the interval-valued fuzzy empty set and 1 to denote the interval-valued fuzzy whole set in a set G, and we define O(x) = [0,0] and l(x) = [1,1]' for all x E G. Definition 2.3. If G* = (V, E) is a graph, then by an interval-valued bifuzzy relation B on a set E we mean an interval-valued bifuzzy set such that JiB(XY) :-:; rmin(jia(x), ItA(y)), for all xy E E. VB(XY) 2': rmax(va(x), VA(Y)) 3 Interval-valued Bifuzzy Graphs Definition 3.1. An interval-valued bijuzzy graph with underlying set V is defined to be a pair G = (A, B) where (i) the functions ita: V -+ D[O, 1] and VA : V -+ D[O, 1] denote the degree of membership and nonmembership ofthe element x E V, respectively such that 0:-:; ItA(X)+VA(X) :-:; 1 for all x E V, (ii) the functions ItB : E ~ V x V -+ D[O, 1] and VB : E ~ V x V -+ D[O, 1] are defined by ItB({X, y}):-:; rmin(lta(x), lta(y)) and VB({X, y}) 2': rmax(va(x),va(y)) such that 0:-:; JiB({X, y}) + VB({X,y}):-:; 1 for all {x,y} E E. We call A the interval-valued bifuzzy vertex set of V, B the interval-valued bifuzzy edge set of G, respectively. Note that B is a symmetric interval-valued bifuzzy relation on A. We use the notation xy for an element of E. Thus, G = (A, B) is an interval-valued bifuzzy graph of G* = (V, E) if ItB(XY) :-:; rmin(lta(x), ItA(y)) and VB(XY) 2': rmax(va(x), VA(y)) for all xy E E.

4 4 Example 3.2. Consider a graph C* = (V, E) such that V = {x,y,z}, E = {xy,yz,zx}. Let A be an interval-valued bifuzzy set of V and let B be an interval-valued bifuzzy set of E ~ V x V defined by x Y z x Y z A =< (------)(------) > [0.1,0.4]' [0.1,0.2]' [0.3,0.4], [0.2,0.5]' [0.3,0.5]' [0.1,0.5], B = (xy yz ZX) (~yz ZX) < [0.06,0.1], [0.05,0.09], [0.06,0.3], [0.1,0.4]' [0.07,0.4]' [0.07,0.4] >. By routine computations, it is easy to see that C = (A, B) is an interval-valued bifuzzy graph of C*. Definition 3.3. Let A = (tta, VA) and A' = (tt~, v~) be interval-valued bifuzzy subsets of VI and V2 and let B = (ttb, VB) and B' = ({L~, vk) be interval-valued bifuzzy subsets of EI and E 2, respectively. The Cartesian product of two interval-valued bifuzzy graphs CI and C2 of the graphs Ci and Ci is denoted by C1 x C 2 = (A X A', B X B') and is defined as follows: (i) (tta x tta)(xi, X2) = rmin(tta(xi), tta(x2)) (VA x V~)(XI' X2) = rmax(va(xi), V~(X2)) for all (Xl, X2) E V, (ii) ({LB x {L~)((X, X2)(X, Y2)) = rmin(tta(x), tt~(x2y2)), (VB x vk)((x, X2)(X, Y2)) = rmax(va(x), Vk(X2Y2)) for all X E VI, for all X2Y2 E E2, (iii) (ttb x tt~)((xi'z)(y),z)) = rmin(ttb(xiyl),tta(z)) (VB x Vk)((XI, Z)(Yl, z)) = rmax(vb(xiyi), v~(z)) for all z E V2, for all XIYl EEl. Proposition 3.4. Let C I and C2 be the interval-valued bijuzzy graphs. Then Cartesian product C I x C2 is an interval-valued bijuzzy graph. (ttb x tt'a)((x, X2)(X, Y2)) (VB x vk)((x, X2)(X, Y2)) rmin(tta(x), tt'a(x2y2)) ::; rmin(tta(x), rmin(tta(x2), tta(y2)) rmin(rmin({la(x), {LA (X2)), rmin({la (X), {LA (Y2))) rmin((tta x {LA) (X, X2), ({LA x {LA) (X, Y2)), rmax(va(x), V~(X2Y2)) :2: rmax(va(x), rmax(v~(x2)' V~(Y2)) rmax(rmax(v A(X), V~ (X2)), rmax(v A(X), V~ (Y2))) rmax((va x v~)(x, X2), (VA x v~)(x, Y2)).

5 5 Let z E 112, XIYl EEl Then (J..tB x J..t~)((Xl' Z)(Yl, z)) This completes the proof. rmin(j..tb(xiyij, J..t~(z)) :::; rmin(rmin(tta(xl)' J..tA(Yl)), J..t~ (z)) rmin(rmin(j..ta(x), J..t~ (z)), rmin(j..ta(yl), J..t~ (z))) rmin((j..ta x J..t~)(XI' z), (J..tA X J..t~)(YI' z)), rmax(vb(xiyi), v~(z)) ;:::: rmax(rmax(va(xi), VA(yIJ), v~(z)) rmax(rmax(va(x), v~(z)), rmax(va(yij, v~(z))) rmax((va x V~)(XI' z), (VA X V~)(YI' z)). Definition 3.5. Let A = (J..tA, VA) and A' = (J..t~, v~) be interval-valued bifuzzy subsets of Vi and V; and let B = (J..tB, VB) and B' = (J..t's, vk) be interval-valued bifuzzy subsets of El and E 2, respectively. The composition of two interval-valued bifuzzy graphs GI and G 2 is denoted by GI [G2] = (A 0 A', B 0 B') and is defined as follows: (i) (J..tA 0 J..t~)(XI,X2) = rmin(j..ta(xl),j..t~(x2)) (VA 0 V~)(XI,X2) = rmax(va(xi), va(x2)) for all (XI,X2) E V, (ii) (J..tB 0 tt~)((x,x2)(x'y2)) = rmin(tta(x),tt's(x2y2)), (VB 0 vk)((x, X2)(X, Y2)) = rmax(va(x), Vk(X2Y2)) for all x E VI, for all X2Y2 E E2, (iii) (J..tB 0 J..t~)((Xl' Z)(Yl, z)) = rmin(j..tb(xiyil, J..tA(z)) (VB 0 Vk)((XI, Z)(YI, z)) = rmax(vb(xiyi), VA(Z)) for all Z E 112, for all XIYI EEl, (iv) (J..tB 0 J..t~)((XI,X2)(YI'Y2)) = rmin(j..t~(x2),j..ta(y2),j..tb(xiyi))' (VB 0 Vk)((XI,X2)(YI,Y2)) = rmax(va(x2),v~(y2),vb(xiyij) for all (XI,X2)(YI,Y2) E EO-E. Proposition 3.6. Let G I and G2 be two inte't"val-valued bijuzzy graphs. Then the composition GI[G2] is an inte't"val-valued bijuzzy graph. o Proof. Let x E Vi and X2Y2 E E2. Then (J..tB 0 J..t~)( (x, X2)(X, Y2)) rmin(j..ta(x), J..t~(X2Y2)) :::; rmin(j..ta(x), rmin(j..t~ (X2)' J..t~ (Y2)) rmin(rmin(j..ta (x), J..t~ (X2)), rmin(j..ta(x), J..t~(Y2))) rmin((j..ta 0 J..t~)(x, X2), (J..tA 0 J..tA)(x, Y2)), (VB 0 vk)((x,x2)(x,y2)) rmax(va(x),v's(x2y2)) ;:::: rmax(va(x), rmax(v~(x2)' V~(Y2)) rmax(rmax(va(x), V~(X2))' rmax(va(x), V~(Y2))) rmax((va 0 v~)(x, X2), (VA 0 v~)(x, Y2)).

6 6 (/lb 0 /l~)((xi' Z)(YI' Z)) (VB X Vk)((XI, Z)(YI, Z)) rmin(/lb(xiyt}, /l~(z)) :::; rmin(rmin(/la(xi), /la(yi)), /l~(z)) rmin(rmin(/la(x), /l~ (z)), rmin(/la (yt}, /l~ (z))) rmin((/la 0 /l~)(xi' z), (/la 0 /l~)(y}, Z)), rmax(vb(xiyi), v~(z)) 2: rmax(rmax(va(xi), VA(YI)), v~(z)) rmax(rmax(va(x), v~(z)), rmax(va (yt}, v~(z))) rmax((va 0 V~)(XI' Z), (VA 0 V~)(YI' z)). Let (Xl, X2)(YI, Y2) E EO - E, so XIYI EEl, X2 -I Y2. Then (/l-b 0 /l~)( (Xl, X2)(YI, Y2)) (VB 0 Vk)((XI, X2)(YI, Y2)) rmin(/l~ (X2), /l~ (Y2), /lb(xiyi)) :::; rmin(/l~ (X2), /l~ (Y2), rmin(/la(xi), /la(yi))) rmin(rmin(/la(xi), /l~ (X2)), rmin(/la(yi), /l~ (Y2))) rmin((/la 0 /l~)(x}, X2), (/la 0 /l~)(yi' Y2)), rmax(v~(x2)' V~(Y2)' VB(XIyt}) 2: rmax(v~(x2) ' V~(Y2) ' rmax(va(xi), VA (YI))) rmax(rmax(v A(XI), V~ (X2)), rmax(v A (YI), V~ (Y2))) rmax((va 0 V~)(XI' X2), (VA 0 V~)(YI' Y2)). This completes the proof. 0 Definition 3.7. Let A = (/la,va) and A' = (/l~,v~) be interval-valued bifuzzy subsets of VI and V2 and let B = (/lb, VB) and B' = (/l~, VB) be interval-valued bifuzzy subsets of EI and E 2, respectively. The union of two interval-valued bifuzzy graphs GI and G2 is denoted by GI U G2 = (A U A', B UB') and is defined as follows: (A) (/la U /l~)(x) = /la(x) if X E Vi n V2, (/la U /l~)(x) = /l~(x) if X E V2 n Vi, (/la U /l~)(x) = rmax(/la(x), /l~(x)) if X E VI n V2 (B) (VA n v~)(x) = VA(X) if X E VI n V2, (VA n v~)(x) = va(x) if X E V2 n Vi, (VA n v~)(x) = rmin(va(x), v~(x)) if x E VI n V2. (C) (/lb U /l~)(xy) = /lb(xy) if xy EEl n E2, (/lb U /l~)(xy) = /l~(xy) if xy E E2 n EI, (/lb U /l~)(xy) = rmax(/lb(xy), /l~(xy)) if xy E EI n E2

7 7 (D) (VB n VB) (xy) = VB(XY) if xy E EI n E2, (VB n VB)(XY) = vb(xy) if xy E E2 n E 1, (VA n VB) (xy) = rmin(vb(xy),vb(xy)) if xy E EI n E2. Proposition 3.8. Let G1 and G2 be two interval-valued bijuzzy graphs. Then the union G1 U G2 is an interval-valued bijuzzy graph. Proof. Let xy E EI n E2 Then IlB u Il's)(xy) (VB n V's) (xy) rma:x(ilb(xy),il's(xy)) ::; rma:x(rmin(ila(x), IlA(Y)), rmin(ll~ (x), Il~ (y))) rmin(rma:x(ila(x), Il~ (x)), rma:x(ila (y), Il~ (y))) rmin((ila U Il~)(x), (IlA U 1l~)(Y)), rmin(vb(xy), VB(XY)) ~ rmin(rma:x(va(x)va(y)), rma:x(v~(x), v~(y))) rma:x(rmin(va(x)v~ (x)), rmin(va(y), v~(y))), rma:x((va(x) n v~)(x), (VA(Y) n v~)(y)). Similarly, we can show that if xy E EI n E2, then (IlA U Il~)(xy) ::; rmin((ila U Il~)(x), (IlA U 1l~)(Y)), (VA n v~)(xy) ~ rma:x((va n v~)(x), (VA n v~)(y)). If xy E E2 n E1, then (IlB U Il's)(xy) ::; rmin((ila U Il~)(x), (IlA U 1l~)(Y)), This completes the proof. (VB n V's) (xy) ~ rma:x((va n v~)(x), (VA n v~)(y)). D Definition 3.9. Let A = (IlA, VA) and A' = (Il~, v~) be interval-valued bifuzzy subsets of Vi and V2 and let B = (IlB, VB) and B' = (Il's, VB) be interval-valued bifuzzy subsets of EI and E2, respectively. The join of two interval-valued bifuzzy graphs G1 and G2 is denoted by G1 + G2 = (A + A', B + B') and is defined as follows: (i) (IlA + Il~)(x) = (IlA + Il~)(x) if x E VI U V2, (VA + v~)(x) = (VA + v~)(x) if x E Vi U \12, (ii) (IlB + Il's)(xy) = (IlB U Il's)(xy) = IlB(XY) (VB + V's) (xy) = (VB n v's)(xy) = VB(XY) if xy E EI U E2,

8 8 (iii) (J.LB + J.L~)(xy) = rmin(j.la(x),j.l~(y)) (VB + vk)(xy) = rmax(va(x), v~(y)) if xy E E'. Proposition Let G] and G 2 be two interval-valued bijuzzy graphs. G] + G2 is an interval-valued bijuzzy graph. Then the join Proof. Let xy E E'. Then (J.LB + J.L~)(xy) (VB + vk)(xy) rmin(j.la(x), J.L~(Y)) ~ rmin((j.la U J.L~)(x), (J.LA U J.L~)(Y)) rmin((j.la + J.L~)(x), (J.LA + J.L~)(Y)), rmax(va(x), v~(y)) ~ rmax((va n v~)(x), (VA n v~)(y)) rmax((va + v~)(x), (VA + v~)(y)). Let xy E E] UE2. Then the result follows from Proposition 3.8. This completes the proof. 0 We formulate the following characterizations. Proposition Let G] and G2 be the interval-valued bijuzzy graphs and let V] n V 2 = 0. Then union G] U G2 is an interval-valued bijuzzy graph ij and only ij G] and G2 are intervalvalued bijuzzy graphs. Proposition Let G] and G2 be two interval-valued bijuzzy graphs and let V] n V 2 = 0. Then the join G] + G2 is an interval-valued bijuzzy graph ij and only ij G] and G2 are interval-valued bijuzzy graphs. Definition Let G = (A, B) be an interval-valued bifuzzy graph. If each vertex has same degree n, then G is called an interval-valued bijuzzy n-regular graph. Definition Let G = (A, B) be an interval-valued bifuzzy graph. The total degree of a vertex x is defined by T deg(x) = ~XYEE J.LB(XY) + J.LA(X) = deg(x) + J.LA(X), T deg(x) = ~XYEEvB(XY) + VA(X) = deg(x) + VA(X). If each vertex of G has total degree n, then G is called an interval-valued bijuzzy n- totally regular graph. Proposition IJ an interval-valued bijuzzy graph G is both regular and totally regular, then A is constant.

9 9 Proof. Let G be a m-regular and interval-valued bifuzzy n-totally regular graphs. Then deg(x) = m and Tdeg(x) = n for all x E V. Now Tdeg(x) deg(x) + {LA (x) m + {LA (x) {LA (x) {LA (x) n n n n-m constant. Likewise, VA(X)= constant. Hence A is a constant function. o We state a characterization without proof. Proposition Let G be an interval-valued bifuzzy graph. Then A is a constant function if and only if the following conditions are equivalent: (1) G is an interval-valued bijuzzy regular graph (2) G is an interval-valued bifuzzy totally regular graph. References [1] M. Akram and W.A. Dudek, Interval-valued fuzzy graphs, Computers and Mathematics with Applications, 61 (2011) [2] M. Akram and KH. Dar, Generalized fuzzy K -algebras, VDM Verlag, 2010, pp.288, ISBN [3] KT. Atanassov, Intuitionistic fuzzy fets: Theory and applications, Studies in fuzziness and soft computing, Heidelberg, New York, Physica-Verl., [4] J.N. Mordeson and C.S. Peng, Operations on fuzzy graphs, Information Sciences 79 (1994) [5] J.N. Mordeson and P.S. Nair, Fuzzy graphs and fuzzy hypergraphs, Physica Verlag, Heidelberg 1998; Second Edition [6] T. Gerstenkorn and J. Manko, Bifuzzy probabilistic sets, Fuzzy Sets and Systems 71 (1995), [7] A. Shannon, KT. Atanassov, A first step to a theory of the intuitionistic fuzzy graphs, Proceeding of FUBEST (D. Lakov, Ed.), Sofia, (1994)

10 10 [8] A. Rosenfeld, Fuzzy graphs, Fuzzy Sets and their Applications( L.A.Zadeh, K.S.Fu, M.Shimura, Eds.), Academic Press, New York, (1975) [9] L.A. Zadeh, Similarity relations and fuzzy orderings, Information Sciences 3(2)(1971) [10] L.A. Zadeh, The concept of a lingusistic variable and its application to appmximate reasoning, Information Sci. 8 (1975)