A graph structure is a useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. In this paper, we apply the concept of bipolar fuzzy sets to graph structures. We introduce certain notions, including bipolar fuzzy graph structure (BFGS), strong bipolar fuzzy graph structure, bipolar fuzzy Ni-cycle, bipolar fuzzy Ni-tree, bipolar fuzzy Ni-cut vertex, and bipolar fuzzy Ni-bridge, and illustrate these notions by several examples. We study ϕ-complement, self-complement, strong self-complement, and totally strong self-complement in bipolar fuzzy graph structures, and we investigate some of their interesting properties.
1. Introduction
Concepts of graph theory have applications in many areas of computer science including data mining, image segmentation, clustering, image capturing, and networking. A graph structure, introduced by Sampathkumar [1], is a generalization of undirected graph which is quite useful in studying some structures including graphs, signed graphs, and graphs in which every edge is labeled or colored. A graph structure helps to study the various relations and the corresponding edges simultaneously.
A fuzzy set, introduced by Zadeh [2], gives the degree of membership of an object in a given set. Zhang [3] initiated the concept of a bipolar fuzzy set as a generalization of a fuzzy set. A bipolar fuzzy set is an extension of fuzzy set whose membership degree range is [-1,1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0,1] of an element indicates that the element somewhat satisfies the property, and the membership degree [-1,0) of an element indicates that the element somewhat satisfies the implicit counterproperty. Kauffman defined in [4] a fuzzy graph. Rosenfeld [5] described the structure of fuzzy graphs obtaining analogs of several graph theoretical concepts. Bhattacharya [6] gave some remarks on fuzzy graphs. Several concepts on fuzzy graphs were introduced by Mordeson et al. [7]. Dinesh [8] introduced the notion of a fuzzy graph structure and discussed some related properties. Akram et al. [9–13] have introduced bipolar fuzzy graphs, regular bipolar fuzzy graphs, irregular bipolar fuzzy graphs, antipodal bipolar fuzzy graphs, and bipolar fuzzy hypergraphs. In this paper, we introduce the certain notions including bipolar fuzzy graph structure (BFGS), strong bipolar fuzzy graph structure, bipolar fuzzy Ni-cycle, bipolar fuzzy Ni-tree, bipolar fuzzy Ni-cut vertex, and bipolar fuzzy Ni-bridge and illustrate these notions by several examples. We present ϕ-complement, self-complement, strong self-complement, and totally strong self-complement in bipolar fuzzy graph structures, and we investigate some of their interesting properties.
We have used standard definitions and terminologies in this paper. For other notations, terminologies, and applications not mentioned in the paper, the readers are referred to [1, 5, 7, 14–18].
2. Preliminaries
In this section, we review some definitions that are necessary for this paper.
A graph structure G∗=(U,E1,E2,…,Ek) consists of a nonempty set U together with relations E1,E2,…,Ek on U, which are mutually disjoint such that each Ei is irreflexive and symmetric. If (u,v)∈Ei for some i,1≤i≤k, we call it an Ei-edge and write it as “uv.” A graph structure G∗=(U,E1,E2,…,Ek) is complete, if (i) each edge Ei,1≤i≤k, appears at least once in G∗; (ii) between each pair of vertices uv in U, uv is an Ei-edge for some i,1≤i≤k. A graph structure G∗=(U,E1,E2,…,Ek) is connected, if the underlying graph is connected. In a graph structure, Ei-path between two vertices u and v, is the path which consists of only Ei-edges for some i, and similarly, Ei-cycle is the cycle which consists of only Ei-edges for some i. A graph structure is a tree if it is connected and contains no cycle or equivalently the underlying graph is a tree. G∗ is an Ei-tree, if the subgraph structure induced by Ei-edges is a tree. Similarly, G∗ is an E1E2⋯Ej-tree, if G∗ is an Ei-tree for each j,1≤j≤k. A graph structure is an Ei-forest, if the subgraph structure induced by Ei-edges is a forest, that is, if it has no Ei-cycles. Let S⊆U; then the subgraph structure 〈S〉 induced by S has vertex set S, where two vertices u and v in 〈S〉 are joined by an Ei-edge, 1≤i≤k, if and only if, they are joined by an Ei-edge in G∗. For some i,1≤i≤k, the Ei-subgraph induced by S is denoted by Ei-〈S〉. It has only those Ei-edges of G∗, joining vertices in S. If T is a subset of edge set in G∗, then subgraph structure 〈T〉 induced by T has the vertex set, “the end vertices in T”, whose edges are those in T. Let G∗=(U1,E1,E2,…,Em) and H∗=(U2,E1′,E2′,…,En′) be graph structures. Then G∗ and H∗ are isomorphic, if (i) m=n, (ii) there exist a bijection f:U1→U2 and a bijection ϕ:{E1,E2,…,En}→{E1′,E2′,…,En′}, say Ei→Ej′, 1≤i,j≤n, such that for all u,v∈U1, uv∈Ei implies that f(u)f(v)∈Ej′.
Two graph structures G∗=(U,E1,E2,…,Ek) and H∗=(U,E1′,E2′,…,Ek′), on the same vertex set U, are identical, if there exists a bijection f:U→U, such that for all u and v in U,uv is an Ei-edge in G∗, then f(u)f(v) is an Ei′-edge in H∗, where 1≤i≤k and Ei≃Ei′∀i. Let ϕ be a permutation on {E1,E2,…,Ek}. Then the ϕ-cyclic complement of G∗, denoted by (G∗)ϕc, is obtained by replacing Ei by ϕ(Ei), 1≤i≤k. Let G∗=(U,E1,E2,…,Ek) be a graph structure and ϕ a permutation on {E1,E2,…,Ek}; then
G∗ is ϕ-self complementary, if G∗ is isomorphic to (G∗)ϕc; the ϕ-cyclic complement of G∗ and G∗ is self-complement, if ϕ≠ identity permutation.
G∗ is strong ϕ-self complementary, if G∗ is identical to (G∗)ϕc; the ϕ-complement of G∗ and G∗ is strong self-complement, if ϕ≠ identity permutation.
Definition 1 (see [2]).
A fuzzy subset μ on a set X is a map μ:X→[0,1]. A fuzzy binary relation on X is a fuzzy subset μ on X×X. By a fuzzy relation we mean a fuzzy binary relation given by μ:X×X→[0,1].
Definition 2 (see [8]).
Let G∗=(U,E1,E2,…,Ek) be a graph structure and let ν,ρ1,ρ2,…,ρk be the fuzzy subsets of U,E1,E2,…,Ek, respectively, such that(1)0≤ρixy≤μx∧μy∀x,y∈U,i=1,2,…,k.Then G=(ν,ρ1,ρ2,…,ρk) is a fuzzy graph structure of G∗.
Definition 3 (see [8]).
Let G=(ν,ρ1,ρ2,…,ρk) be a fuzzy graph structure of a graph structure G∗=(U,E1,E2,…,Ek). Then F=(ν,τ1,τ2,…,τk) is a partial fuzzy spanning subgraph structure of G if τi⊆ρi for i=1,2,…,k.
Definition 4 (see [8]).
Let G∗ be a graph structure and let G be a fuzzy graph structure of G∗. If xy∈supp(ρi), then “xy” is said to be a ρi-edge of G.
Definition 5 (see [8]).
The strength of a ρi-path x0x1⋯xn of a fuzzy graph structure G is ⋀j=1nρi(xj-1xj) for i=1,2,…,k.
Definition 6 (see [8]).
In a fuzzy graph structure G,ρi2(xy)=ρi∘ρi(xy)=⋁z{ρi(xz)∧ρi(zy)}, ρij(xy)=(ρij-1∘ρi)(xy)=⋁z{ρij-1(xz)∧ρi(zy)}, j=2,3,…,m, for any m≥2. Also ρi∞xy=⋁{ρij(xy),j=1,2,…}.
Definition 7 (see [8]).
Let xy be a ρi-edge of G=(ν,ρ1,ρ2,…,ρn). Let (ν,ρ1′,ρ2′,…,ρn′) be a partial fuzzy spanning subgraph structure obtained by deleting “xy” with ρi′(xy)=0 and ρi′(x1y1)=ρi(x1y1)∀ρi-edges (x1y1) other than (x,y). If ρi∞(uv)>ρi′∞(uv) for some uv∈ supp(ρi), then xy is a ρi-bridge.
Definition 8 (see [8]).
Let G′=(ν,ρ1′,ρ2′,…,ρn′) be the partial fuzzy subgraph structure obtained by deleting vertex w of G, that is, ν′(w)=0 and ν′(v)=ν(v)∀v≠w,ρi′(vw)=0∀v∈Ur and ρi′(uv)=ρi(uv)∀uv≠wv,i=1,2,…,k. Then a vertex w of G is a ρi-cut vertex if ρi∞(uv)>ρi′∞(uv) for some u,v with u,v≠w.
Definition 9 (see [8]).
G=(ν,ρ1,ρ2,…,ρk) is a ρi-cycle if and only if (suppν,suppρ1,suppρ2,…,supp(ρk)) is a Ei-cycle.
Definition 10 (see [8]).
G=(ν,ρ1,ρ2,…,ρk) is a fuzzy ρi-cycle if and only if (suppν,suppρ1,suppρ2,…,supp(ρk)) is an Ei-cycle and there exists no unique “xy” in supp(ρi) such that ρi(xy)=⋀{ρi(uv)∣uv∈supp(ρi)}.
Definition 11 (see [8]).
G=(ν,ρ1,ρ2,…,ρk) is a fuzzy ρi-tree if it has a partial fuzzy spanning subgraph structure, Fi=(ν,τ1,τ2,…,τk), which is a τi-tree where for all ρi-edges not in Fi,ρi(xy)<τi∞(xy).
Definition 12 (see [8]).
Let G∗=(U,E1,E2,…,Ek) be a graph structure and let ν,ρ1,ρ2,…,ρk be the fuzzy subsets of U,E1,E2,…,Ek, respectively, such that(2)0≤ρixy≤μx∧μy∀x,y∈V,i=1,2,…,k.Then G=(ν,ρ1,ρ2,…,ρk) is a fuzzy graph structure of G∗.
Definition 13 (see [3]).
Let X be a nonempty set. A bipolar fuzzy set B in X is an object having the form (3)B=x,μBPx,μBNx∣x∈X,where μBP:X→[0,1] and μBN:X→[-1,0] are mappings.
We use the positive membership degree μBP(x) to denote the satisfaction degree of an element x to the property corresponding to a bipolar fuzzy set B and the negative membership degree μBN(x) to denote the satisfaction degree of an element x to some implicit counterproperty corresponding to a bipolar fuzzy set B. If μBP(x)≠0 and μBN(x)=0, it is the situation that x is regarded as having only positive satisfaction for B. If μBP(x)=0 and μBN(x)≠0, it is the situation that x does not satisfy the property of B but somewhat satisfies the counter property of B. It is possible for an element x to be such that μBP(x)≠0 and μBN(x)≠0 when the membership function of the property overlaps that of its counterproperty over some portion of X.
For the sake of simplicity, we will use the symbol B=(μBP,μBN) for the bipolar fuzzy set: (4)B=x,μBPx,μBNx∣x∈X.
Definition 14 (see [3]).
Let X be a nonempty set. Then we call a mapping A=(μAP,μAN):X×X→[0,1]×[-1,0] a bipolar fuzzy relation on X such that μAP(x,y)∈[0,1] and μAN(x,y)∈[-1,0].
Definition 15 (see [9]).
A bipolar fuzzy graph G=(V,A,B) is a nonempty set V together with a pair of functions A=(μAP,μAN):V→[0,1]×[-1,0] and B=(μBP,μBN):V×V→[0,1]×[-1,0] such that for all x,y∈V, (5)μBPx,y≤minμAPx,μAPy,μBNx,y≥maxμANx,μANy.
Notice that μBP(x,y)>0, μBN(x,y)<0 for (x,y)∈V×V, μBP(x,y)=μBN(x,y)=0 for (x,y)∉V×V, and B is symmetric relation.
3. Bipolar Fuzzy Graph StructuresDefinition 16.
Gˇb=(M,N1,N2,…,Nn) is called a bipolar fuzzy graph structure (BFGS) of a graph structure (GS) G∗=(U,E1,E2,…,En) if M=(μMP,μMN) is a bipolar fuzzy set on U and for each i=1,2,…,n; Ni=(μNiP,μNiN) is a bipolar fuzzy set on Ei such that (6)μNiPxy≤μMPx∧μMPy,μNiNxy≥μMNx∨μMNy∀xy∈Ei⊂U×U.Note that μNiP(xy)=0=μNiN(xy) for all xy∈U×U-Ei and 0<μNiP(xy)≤1, -1≤μNiN(xy)<0∀xy∈Ei, where U and Ei(i=1,2,…,n) are called underlying vertex set and underlying i-edge set of Gˇb, respectively.
Definition 17.
Let Gˇb=(M,N1,N2,…,Nn) be a bipolar fuzzy graph structure of a graph structure G∗=(U,E1,E2,…,En). If Hˇb=(M′,N1′,N2′,…,Nn′) is a bipolar fuzzy graph structure of G∗ such that (7)μM′Px≤μMPx,μM′Nx≥μMNx∀x∈U,μNi′Pxy≤μNiPxy,μNi′Nxy≥μNiNxy∀xy∈Ei,i=1,2,…,n,then Hˇb is called a bipolar fuzzy subgraph structure of BFGS Gˇb.
BFGS Hˇb=(M′,N1′,N2′,…,Nn′) is a bipolar fuzzy induced subgraph structure of Gˇb=(M,N1,N2,…,Nn), by a subset W of U if (8)μM′Px=μMPx,μM′Nx=μMNx∀x∈W,μNi′Pxy=μNiPxy,μNi′Nxy=μNiNxy∀x,y∈W,i=1,2,…,n.Similarly, BFGS Hˇb is a bipolar fuzzy spanning subgraph structure of Gˇb if M′=M and (9)μNi′P≤μNiP,μNi′N≥μNiN,i=1,2,…,n.
Example 18.
Consider a graph structure G∗=(U,E1,E2) such that U={a1,a2,a3,a4},E1={a1a2,a2a4}, and E2={a3a4,a1a4}.
(i) Let M,N1, and N2 be bipolar fuzzy subsets of U,E1, and E2, respectively, such that (10)M=a1,0.5,-0.2,a2,0.7,-0.3,a3,0.4,-0.3,a4,0.7,-0.3,N1=a1a2,0.5,-0.2,a2a4,0.7,-0.3,N2=a3a4,0.3,-0.2,a1a4,0.3,-0.1.Then, by direct calculations, it is easy to see that Gˇb=(M,N1,N2) is a BFGS of G∗ as shown in Figure 1.
(ii) Consider M1={(a1,0.4,-0.1),(a2,0.5,-0.3),(a3,0.4,-0.2),(a4,0.1,-0.3)}, N11={(a1a2,0.4,-0.1),(a2a4,0.1,-0.2)}, and N12={(a3a4,0.1,-0.2),(a1a4,0.1,-0.0)}. Then, by routine calculations, it is easy to see that Kˇb=(M1,N11,N12) is the bipolar fuzzy subgraph structure of Gˇb as shown in Figure 2.
Let Gˇb=(M,N1,N2,…,Nn) be a bipolar fuzzy graph structure of a graph structure G∗=(U,E1,E2,…,En). Then xy∈Ei is called a bipolar fuzzy Ni-edge or simply Ni-edge, if(11)μNiPxy>0 orμNiNxy<0.Then support of Ni, i=1,2,…,n, consequently, is (12)suppNi=xy∈Ei:μNiPxy>0,μNiNxy<0.
Definition 20.
Ni-path in a BFGS Gˇb=(M,N1,N2,…,Nn) of a graph structure G∗=(U,E1,E2,…,En) is a sequence a1,a2,…,am of distinct vertices (except the choice am=a1)in U, such that aj-1aj is a bipolar fuzzy Ni-edge for all j=2,3,…,m.
Definition 21.
A BFGS Gˇb=(M,N1,N2,…,Nn) with underlying vertex set U is said to be Ni-strong for some i∈{1,2,3,…,n} if for all xy∈supp(Ni)(13)μNiPxy=μMPx∧μMPy,μNiNxy=μMNx∨μMNy.A BFGS Gˇb=(M,N1,N2,…,Nn) is said to be strong if it is Ni-strong BFGS for all i∈{1,2,3,…,n}.
Example 22.
Consider BFGS Gˇb=(M,N1,N2) as shown in Figure 3.
Then Gˇb is a strong BFGS since it is both N1- and N2-strong.
BFGS Gˇb=(M,N1,N2).
Definition 23.
A BFGS Gˇb=(M,N1,N2,…,Nn) with underlying vertex set U is said to be complete or N1N2⋯Nn-complete, if the following are true:
Gˇb a is strong BFGS.
supp(Ni)≠∅∀i=1,2,3,…,n.
For each pair of vertices x,y∈U,xy is an Ni-edge for some i.
Example 24.
Let Gˇb=(M,N1,N2) be BFGS of graph structure G∗=(U,E1,E2) such that U={a1,a2,a3},E1={a2a3}, and E2={a1a2,a1a3} as shown in Figure 4. By routine calculations, it is easy to see that Gˇb is a strong BFGS.
Moreover, supp(N1)≠∅,supp(N2)≠∅, and every pair of vertices belonging to U is either an N1-edge or an N2-edge. So Gˇb is a complete BFGS, that is, N1N2-complete BFGS.
Gˇb=(M,N1,N2).
Definition 25.
Let Gˇb=(M,N1,N2,…,Nn) be a BFGS with underlying vertex set U. Then positive and negative strengths of a Ni-path “PNi=a1a2⋯am” are called gain and loss of that Ni-path and denoted by G.PNi and L.PNi, respectively, such that (14)G.PNi=⋀j=2mμNiPaj-1aj,L.PNi=⋁j=2mμNiNaj-1aj.
Example 26.
Consider a BFGS Gˇb=(M,N1,N2) as shown in Figure 4. We note that PN2=a1a3a4a1 is an N2-path. So G.PN2=μN2P(a3a1)∧μN2P(a1a2)=0.5∧0.4=0.4. Consider (15)L.PN2=μN2Na3a1∨μN2Na1a2=-0.4∨-0.4=-0.4=0.4.
Definition 27.
Let Gˇb=(M,N1,N2,…,Nn) be a BFGS with underlying vertex set U. Then
Ni-gain of connectedness between x and y is defined by μNi∞,+(xy)=⋁j≥1{μNij,+(xy)}, such that μNij,+(xy)=(μNij-1,+∘μNi1,+)(xy) for j≥2 and μNi2,+(xy)=(μNi1,+∘μNi1,+)(xy)=⋁z{μNi1,+(xz)∧μNi1,+(zy)}, where μNi1,+=μNiP,∀i.
Ni-loss of connectedness between x and y is defined by μNi∞,-(xy)=⋁j≥1{μNij,-(xy)}, such that μNij,-(xy)=(μNij-1,-∘μNi1,-)(xy) for j≥2 and μNi2,-(xy)=(μNi1,-∘μNi1,-)(xy)=⋁z{μNi1,-(xz)∧μNi1,-(zy)}, where μNi1,-=μNiN,∀i.
Example 28.
Let Gˇb=(M,N1,N2) be BFGS of graph structure G=(U,E1,E2) such that U={a1,a2,a3,a4,a5,a6}, E1={a1a6,a2a3,a2a5,a3a4,a4a5}, and E2={a1a3,a1a2,a4a6,a5a6}, as is shown in Figure 5.
Since μN11,+(a2a3)=0.3, μN11,+(a2a4)=0.0, μN11,+(a2a5)=0.4, μN11,+(a3a4)=0.5, μN11,+(a5a3)=0.0, μN11,+(a4a5)=0.3, and μN11,+(a1a6)=0.3, therefore(16)μN12,+a2a3=μ1,+nN1∘μN11,+a2a3=μN11,+a2a4∧μN11,+a4a3∨μN11,+a2a5∧μN11,+a5a3=0.0∧0.5∨0.4∧0.0=0,μN12,+a2a4=μ1,+nN1∘μN11,+a2a4=μN12,+a2a3∧μN11,+a3a4∨μN11,+a2a5∧μN11,+a5a4=0.3∧0.5∨0.4∧0.3=0.3,μN12,+a2a5=μ1,+nN1∘μN11,+a2a5=μN11,+a2a3∧μN11,+a3a5∨μN11,+a2a4∧μN11,+a4a5=0.3∧0.0∨0.0∧0.3=0,μN12,+a3a4=μ1,+nN1∘μN11,+a3a4=μN11,+a3a2∧μN11,+a2a4∨μN11,+a3a5∧μN11,+a5a4=0.3∧0.0∨0.0∧0.3=0,μN12,+a3a5=μ1,+nN1∘μN11,+a3a5=μN11,+a3a2∧μN11,+a2a5∨μN11,+a3a4∧μN11,+a4a5=0.3∧0.4∨0.5∧0.3=0.3,μN12,+a4a5=μ1,+nN1∘μN11,+a4a5=μN11,+a4a2∧μN11,+a2a5∨μN11,+a4a3∧μN11,+a3a5=0.0∧0.4∨0.5∧0.0=0,μN12,+a1a6=μ1,+nN1∘μN11,+a1a6=0,μN13,+a2a3=μ2,+nN1∘μN11,+a2a3=μN12,+a2a4∧μN11,+a4a3∨μN12,+a2a5∧μN11,+a5a3=0.3∧0.5∨0.0∧0.0=0.3,μN13,+a2a4=μ2,+nN1∘μN11,+a2a4=μN12,+a2a3∧μN11,+a3a4∨μN12,+a2a5∧μN11,+a5a4=0.0∧0.5∨0.0∧0.3=0.0,μN13,+a2a5=μ2,+nN1∘μN11,+a2a5=μN12,+a2a3∧μN11,+a3a5∨μN12,+a2a4∧μN11,+a4a5=0.0∧0.0∨0.3∧0.3=0.3,μN13,+a3a4=μ2,+nN1∘μN11,+a3a4=μN12,+a3a2∧μN11,+a2a4∨μN12,+a3a5∧μN11,+a5a4=0.0∧0.0∨0.3∧0.3=0.3,μN13,+a3a5=μ2,+nN1∘μN11,+a3a5=μN12,+a3a2∧μN11,+a2a5∨μN12,+a3a4∧μN11,+a4a5=0.0∧0.4∨0.0∧0.3=0.0,μN13,+a4a5=μ2,+nN1∘μN11,+a4a5=μN12,+a4a2∧μN11,+a2a5∨μN12,+a4a3∧μN11,+a3a5=0.3∧0.4∨0.0∧0.0=0.3,μN13,+a1a6=μ2,+nN1∘μN11,+a1a6=0.
Similarly,(17)μN14,+a2a3=μ1,+nN1∘μN11,+a2a3=0,μN14,+a2a4=μ1,+nN1∘μN11,+a2a4=0.3,μN14,+a2a5=μ1,+nN1∘μN11,+a2a5=0,μN14,+a3a4=μ1,+nN1∘μN11,+a3a4=0,μN14,+a3a5=μ1,+nN1∘μN11,+a3a5=0.3,μN14,+a4a5=μ1,+nN1∘μN11,+a4a5=0,μN14,+a1a6=μ1,+nN1∘μN11,+a1a6=0.This implies that(18)μN1∞,+a2a3=∨0.3,0.0,0.3,0.0=0.3,μN1∞,+a2a4=∨0.0,0.3,0.0,0.3=0.3,μN1∞,+a2a5=∨0.4,0.0,0.3,0.0=0.4,μN1∞,+a3a4=∨0.5,0.0,0.3,0.0=0.5,μN1∞,+a3a5=∨0.0,0.3,0.0,0.3=0.3,μN1∞,+a4a5=∨0.3,0.0,0.3,0.0=0.3,μN1∞,+a1a6=∨0.3,0.0,0.0,0.0=0.3.Since(19)μN11,-a2a3=0.5,μN11,-a2a4=0.0,μN11,-a2a5=0.4,μN11,-a3a4=0.7,μN11,-a5a3=0.0,μN11,-a4a5=0.3,μN11,-a1a6=0.2,we have(20)μN12,-a2a3=μ1,-nN1∘μN11,-a2a3=μN11,-a2a4∧μN11,-a4a3∨μN11,-a2a5∧μN11,-a5a3=0.0∧0.7∨0.4∧0.0=0.0,μN12,-a2a4=μ1,-nN1∘μN11,-a2a4=μN12,-a2a3∧μN11,-a3a4∨μN11,-a2a5∧μN11,-a5a4=0.5∧0.7∨0.4∧0.3=0.5,μN12,-a2a5=μ1,-nN1∘μN11,-a2a5=μN11,-a2a3∧μN11,-a3a5∨μN11,-a2a4∧μN11,-a4a5=0.5∧0.0∨0.0∧0.3=0.0,μN12,-a3a4=μ1,-nN1∘μN11,-a3a4=μN11,-a3a2∧μN11,-a2a4∨μN11,-a3a5∧μN11,-a5a4=0.5∧0.0∨0.0∧0.3=0.0,μN12,-a3a5=μ1,-nN1∘μN11,-a3a5=μN11,-a3a2∧μN11,-a2a5∨μN11,-a3a4∧μN11,-a4a5=0.5∧0.4∨0.7∧0.3=0.4,μN12,-a4a5=μ1,-nN1∘μN11,-a4a5=μN11,-a4a2∧μN11,-a2a5∨μN11,-a4a3∧μN11,-a3a5=0.0∧0.4∨0.7∧0.0=0.0,μN12,-a1a6=μ1,-nN1∘μN11,-a1a6=0,μN13,-a2a3=μ2,-nN1∘μN11,-a2a3=μN12,-a2a4∧μN11,-a4a3∨μN12,-a2a5∧μN11,-a5a3=0.5∧0.7∨0.0∧0.0=0.5,μN13,-a2a4=μ2,-nN1∘μN11,-a2a4=μN12,-a2a3∧μN11,-a3a4∨μN12,-a2a5∧μN11,-a5a4=0.0∧0.7∨0.0∧0.3=0.0,μN13,-a2a5=μ2,-nN1∘μN11,-a2a5=μN12,-a2a3∧μN11,-a3a5∨μN12,-a2a4∧μN11,-a4a5=0.0∧0.0∨0.5∧0.3=0.3,μN13,-a3a4=μ2,-nN1∘μN11,-a3a4=μN12,-a3a2∧μN11,-a2a4∨μN12,-a3a5∧μN11,-a5a4=0.0∧0.0∨0.4∧0.3=0.3,μN13,-a3a5=μ2,-nN1∘μN11,-a3a5=μN12,-a3a2∧μN11,-a2a5∨μN12,-a3a4∧μN11,-a4a5=0.0∧0.4∨0.0∧0.3=0.0,μN13,-a4a5=μ2,-nN1∘μN11,-a4a5=μN12,-a4a2∧μN11,-a2a5∨μN12,-a4a3∧μN11,-a3a5=0.5∧0.4∨0.0∧0.0=0.4,μN13,-a1a6=μ2,-nN1∘μN11,-a1a6=0.Similarly,(21)μN14,-a2a3=μ1,-nN1∘μN11,-a2a3=0,μN14,-a2a4=μ1,-nN1∘μN11,-a2a4=0.5,μN14,-a2a5=μ1,-nN1∘μN11,-a2a5=0,μN14,-a3a4=μ1,-nN1∘μN11,-a3a4=0,μN14,-a3a5=μ1,-nN1∘μN11,-a3a5=0.4,μN14,-a4a5=μ1,-nN1∘μN11,-a4a5=0,μN14,-a1a6=μ1,-nN1∘μN11,-a1a6=0.This implies that(22)μN1∞,-a2a3=∨0.5,0.0,0.5,0.0=0.5,μN1∞,-a2a4=∨0.0,0.5,0.0,0.5=0.5,μN1∞,-a2a5=∨0.4,0.0,0.3,0.0=0.4,μN1∞,-a3a4=∨0.7,0.0,0.3,0.0=0.7,μN1∞,-a3a5=∨0.0,0.4,0.0,0.4=0.4,μN1∞,-a4a5=∨0.3,0.0,0.4,0.0=0.4,μN1∞,-a1a6=∨0.3,0.0,0.0,0.0=0.2.For all the remaining pairs of vertices, N1-loss and N1-gain of connectedness are zero.
Gˇb=(M,N1,N2).
Definition 29.
A BFGS Gˇb=(M,N1,N2,…,Nn) of a graph structure G∗=(U,E1,E2,…,En) is an Ni-cycle if (suppM,suppN1,suppN2,…,supp(Nn)) is an Ei-cycle.
Definition 30.
A BFGS Gˇb=(M,N1,N2,…,Nn) of a graph structure G∗=(U,E1,E2,…,En) is a bipolar fuzzy Ni-cycle for some i if
Gˇb is an Ni-cycle;
there is no unique Ni-edge uv in Gˇb such that μNiP(uv)=min{μNiP(xy):xy∈Ei=supp(Ni)} or μNiN(uv)=max{μNiN(xy):xy∈Ei=supp(Ni)}.
Example 31.
Consider BFGS Gˇb=(M,N1,N2) as shown in Figure 3. Then Gˇb is an N1-cycle as well as bipolar fuzzy N1-cycle, since (suppM,suppN1,supp(N2)) is an E1-cycle and there are two N1-edges with minimum positive degree and more than one N1-edge with maximum negative degree of all N1-edges.
Definition 32.
Let Gˇb=(M,N1,N2,…,Nn) be a BFGS of a graph structure G∗=(U,E1,E2,…,En) and x a vertex of Gˇb. Let (M′,N1′,N2′,…,Nn′) be a bipolar fuzy subgraph structure of Gˇb induced by U∖{x} such that (23)μM′Px=0=μM′Nx,μNi′Pxv=0=μNi′Nxv∀edgesxv∈Gˇb,μM′Pv=μMPv,μM′Nv=μMNv,∀v≠x,μNi′Puv=μNiPuv,μNi′Nuv=μNiNuv∀i, such that u≠x,v≠x.Then x is a bipolar fuzzy Ni-cut vertex for some i, if(24)μNi∞,+uv>μNi′∞,+uv,μNi∞,-uv>μNi′∞,-uvfor some u,v∈U∖x.And, x is an Ni-P bipolar fuzzy cut vertex if only the first condition holds and a Ni-N bipolar fuzzy cut vertex if only the second condition holds.
Example 33.
Consider BFSG Gˇb=(M,N1,N2) as considered in Example 28 and shown in Figure 5; after deleting vertex a2, the resulting bipolar fuzzy subgraph structure will be as shown in Figure 6.
Then a2 is a bipolar fuzzy N1-N cut vertex since(25)μN1∞,-a3a4=0.7=μN1′∞,-a3a4,μN1∞,-a3a5=0.4>0.3=μN1∞,-a3a5,μN1∞,-a4a5=0.4>0.3=μN1′∞,-a4a5,μN1∞,-a1a6=0.2=μN1∞,-a1a6.
Let Gˇb=(M,N1,N2,…,Nn) be a BFGS of a graph structure G∗=(U,E1,E2,…,En) and let xy be an Ni-edge. Let (M,N1′,N2′,…,Nn′) be a bipolar fuzzy spanning subgraph structure of Gˇb, obtained by taking (26)μNi′Pxy=0=μNi′Nxy,μNi′Puv=μNiPuv,μNi′Nuv=μNiNuv∀edgesuv≠xy.Then xy is a bipolar fuzzy Ni-bridge if (27)μNi∞,+uv>μNi′∞,+uv,μNi∞,-uv>μNi′∞,-uvfor some u,v∈U.Edge xy is an Ni-P bipolar fuzzy bridge if only the first condition holds and an Ni-N bipolar fuzzy bridge if only the second condition holds.
Example 35.
Consider the BFGS Gˇb=(M,N1,N2) as shown in Figure 6 and let Gˇb′=(M,N1′,N2′) be bipolar fuzzy spanning subgraph structure of Gˇb obtained by deleting N1-edge (a2a5). Then a2a5 is a bipolar fuzzy N1-bridge, since μN1∞,+(a2a5)=0.4>0.3=μN1′∞,+(a2a5) and μN1∞,-(a2a5)=0.4>0.3=μN1′∞,-(a2a5), and also abipolar fuzzy N1-N bridge, since μN1∞,-(a3a5)=0.4>0.3=μN1′∞,-(a3a5) and μN1∞,-(a4a5)=0.4>0.3=μN1′∞,-(a4a5).
Definition 36.
A BFGS Gˇb=(M,N1,N2,…,Nn) of a graph structure G∗=(U,E1,E2,…,En) is an Ni-tree if (suppA,suppN1,suppN2,…,supp(Nn)) is an Ei-tree. In other words, Gˇb is an Ni-tree if a subgraph of Gˇb, induced by supp(Ni), forms a tree.
Definition 37.
A BFGS Gˇb=(M,N1,N2,…,Nn) of a graph structure G∗=(U,E1,E2,…,En) is a bipolar fuzzy Ni-tree if Gˇb has a bipolar fuzzy spanning subgraph structure Hˇb=(A,C1,C2,…,Cn) such that Hˇb is a Ci-tree and μNiP(xy)<μCi∞,+(xy) and μNiNxy<μCi∞,-(xy)∀Ni-edges not in Hˇb.
In more concerned view, Gˇb is a bipolar fuzzy Ni-P tree if only the first condition holds and a bipolar fuzzy Ni-N tree if only the second condition holds.
Example 38.
Consider BFGS Gˇb=(M,N1,N2) as shown in Figure 7, which is an N2-tree. It is not an N1-tree but a bipolar fuzzy N1-tree since it has a bipolar fuzzy spanning subgraph structure (M,N1′,N2′) as an N1-tree, which is obtained by deleting N1-edge a2a5 from Gˇb and(28)μN1Pa2a5=0.2<0.3=μN1′∞,+a2a5,μN1Na2a5=0.1<0.3=μN1′∞,-a2a5.
Gˇb=(M,N1,N2).
Definition 39.
A BFGS Gˇs1=(M1,N11,N12,…,N1n) of graph structure G1∗=(U1,E11,E12,…,E1n) is isomorphic to a BFGS Gˇs2=(M2,N21,N22,…,N2n) of G2∗=(U2,E21,E22,…,E2n) if there exists a bijection f:U1→U2 and a permutation ϕ on the set {1,2,…,n} such that(29)μM1Pu1=μM2Pfu1,μM1Nu1=μM2Nfu1∀u1∈U1and for ϕ(i)=j(30)μN1iPu1u2=μN2jPfu1fu2,μN1iNu1u2=μN2jNfu1fu2∀u1u2∈E1i,i=1,2,…,n.
Example 40.
Let Gˇb1=(M,N1,N2) and Gˇb2=(M′,N1′,N2′) be two BFGSs of graph structures G1∗=(U,E1,E2) and G2∗=(U′,E1′,E2′), respectively, as shown in Figure 8.
Here Gˇb1 is isomorphic (not identical) to Gˇb2 under the mapping f:U→U′, defined by f(a1)=b1, f(a2)=b2, and f(a3)=b3, and a permutation ϕ given by ϕ(1)=2, ϕ(2)=1, such that(31)μMPai=μM′Pfai,μMNai=μM′Nfai∀ai∈U,μNkPaiaj=μNϕkPfaifaj,μNkNaiaj=μNϕkNfaifaj∀aiaj∈Ek,k=1,2.
Isomorphic bipolar fuzzy graph structures.
Definition 41.
A BFGS Gˇs1=(M1,N11,N12,…,N1n) of GS G1∗=(U,E11,E12,…,E1n) is identical to a BFGS Gˇs2=(M2,N21,N22,…,N2n) of GS G2∗=(U,E21,E22,…,E2n) if there exist a bijection f:U→U, such that(32)μM1Pu=μM2Pfu,μM1Nu=μM2Nfu∀u∈U,μN1iPu1u2=μN2iPfu1fu2,μN1iNu1u2=μN2iNfu1fu2∀u1u2∈E1i,i=1,2,…,n.
Example 42.
Let Gˇb1=(M,N1,N2) and Gˇb2=(M′,N1′,N2′) be two BFGSs of graph structures G1∗=(U,E1,E2) and G2∗=(U′,E1′,E2′), respectively, as shown in Figure 9.
Here Gˇb1 is identical with Gˇb2 under the mapping f:U→U′, defined by f(a1)=b6,f(a2)=b2,f(a3)=b4,f(a4)=b5,f(a5)=b1, and f(a6)=b3, such that(33)μMPai=μM′Pfai,μMNai=μM′Nfai∀ai∈U,μNkPaiaj=μNk′Pfaifaj,μNkNaiaj=μNk′Nfaifaj∀aiaj∈Ek,k=1,2.
Identical bipolar fuzzy graph structures.
Definition 43.
Let Gˇb=(M,N1,N2,…,Nn) be a BFGS of a GS G∗=(U,E1,E2,…,En). Let ϕ be any permutation on the set {E1,E2,…,En} and the corresponding permutation on {N1,N2,…,Nn}; that is, ϕ(Ni)=Nj if and only if ϕ(Ei)=Ej∀i.
If xy∈Nr for some r and (34)μNiϕPxy=μMPx∧μMPy-⋁j≠iμϕNjPxy,μNiϕNxy=μMNx∨μMNy-⋀j≠iμϕNjNxy,i=1,2,…,n,then xy∈Bmϕ, while m is chosen such that μNmϕP(xy)≥μNiϕP(xy) and μNmϕN(xy)≤μNiϕN(xy)∀i.
And BFGS (M,N1ϕ,N2ϕ,…,Nnϕ), denoted by Gˇbϕc, is called the ϕ-complement of BFGS Gˇb.
Example 44.
Let M={(a1,0.3,-0.7), (a2,0.5,-0.4), (a3,0.7,-0.3)}, N1={(a1a3,0.3,-0.3),(a2a3,0.5,-0.3)}, and N2={(a1a2,0.3,-0.4)} be bipolar fuzzy subsets of U,E1, and E2, respectively, so that Gˇb=(M,N1,N2) is a BFGS of graph structure G∗=(U,E1,E2). Let ϕ be a permutation on the set {N1,N2} such that ϕ(N1)=N2 and ϕ(N2)=N1.
Now for a2a3∈N1,(35)μN1ϕPa2a3=μMPa2∧μMPa3-⋁j≠1μϕNjPa2a3=0.5∧0.7-μϕN2Pa2a3=0.5-μN1Pa2a3=0.5-0.5=0,μN1ϕNa2a3=μMNa2∨μMNa3-⋀j≠1μϕNjNa2a3=-0.4∨-0.3-μϕN2Na2a3=-0.3-μN1Na2a3=-0.3+0.3=0,μN2ϕPa2a3=μMPa2∧μMPa3-⋁j≠2μϕNjPa2a3=0.5∧0.7-μϕN1Pa2a3=0.5-μN2Pa2a3=0.5-0=0.5,μN2ϕNa2a3=μMNa2∨μMNa3-⋀j≠2μϕNjNa2a3=-0.4∨-0.3-μϕN1Na2a3=-0.3-μN2Na2a3=-0.3-0=-0.3.Clearly, μϕN2P(a2a3)=0.5>0=μϕN1P(a2a3) and μϕN2N(a2a3)=-0.3<0=μϕN1N(a2a3). So a2a3∈N2ϕ.
Similarly for a1a3∈N1,μN1ϕP(a1a3)=0,μN1ϕN(a1a3)=0,μN2ϕP(a1a3)=0.3, and μN2ϕNa1a3=-0.3.
⇒μN2ϕP(a1a3)=0.3>0=μN1ϕP(a1a3) and μN2ϕN(a1a3)=-0.3<0=μN1ϕN(a1a3). So a1a3∈N2ϕ.
And for a1a2∈N2,μN1ϕP(a1a2)=0.3,μN1ϕN(a1a2)=-0.4,μN2ϕP(a1a2)=0, and μN2ϕN(a1a2)=0.
⇒μN1ϕP(a1a2)=0.3>0=μN2ϕP(a1a2) and μN1ϕN(a1a2)=-0.4<0=μN2ϕN(a1a2). So a1a2∈N1ϕ.
This implies that(36)N1ϕ=a1a2,0.3,-0.4,N2ϕ=a2a3,0.5,-0.3,a1a3,0.3,-0.3and Gˇbϕc=(M,N1ϕ,N2ϕ) is the ϕ-complement of Gˇb.
Theorem 45.
A ϕ-complement of a bipolar fuzzy graph structure is always a strong BFGS. Moreover, if ϕ(i)=r for r,i∈{1,2,…,n}, then all Nr-edges in BFGS Gˇb=(M,N1,N2,…,Nn) become Biϕ-edges in Gˇbϕc=(A,B1ϕ,B2ϕ,…,Bnϕ).
Proof.
From the definition of ϕ-complement Gˇbϕc,(37)μNiϕPxy=μMPx∧μMPy-⋁j≠iμϕNjPxy,(38)μNiϕNxy=μMNx∨μMNy-⋀j≠iμϕNjNxy,for i=1,2,…,n.
Let us consider expression (37) first.
Since μMN(x)∨μMN(y)≤0 and ⋀j≠iμϕNjN(xy)≤0, we can write (39)μNiϕNxy=-μMNx∨μMNy+⋀j≠iμϕNjNxy.Also from the definition of a BFGS μNjNxy≥μMNx∨μMNy∀Nj(40)⟹⋀j≠iμϕNjNxy≥μMNx∨μMNy⟹⋀j≠iμϕNjNxy≤μMNx∨μMNy⟹-μMNx∨μMNy+⋀j≠iμϕNjNxy≤0.Therefore, μNiϕN(xy)≤0∀i.
Now a requirement is minimum value of μNiϕN(xy). Since μNiϕN(xy)≤0, that is why it is minimum when its positive part ⋀j≠iμϕNjNxy is zero. And ⋀j≠iμϕNjNxy=0 when ϕNi=Nr and xy is an Nr-edge. So(41)μNiϕNxy=μMNx∨μMNy,for xy∈Nr,ϕNi=Nr.Similarly for expression (38), a requirement is maximum value of μNiϕPxy. Since μMPx∧μMPy≥0, ⋁j≠iμϕNjPxy≥0 and μNjPxy≤μMPx∧μMPy∀Nj(42)⟹⋁j≠iμϕNjPxy≤μMPx∧μMPy⟹μMPx∧μMPy-⋁j≠iμϕNjPxy≥0.Therefore, μNiϕP(xy)≥0∀i.
Now μNiϕP(xy) will be maximum when its negative part [-⋁j≠iμϕNjP(xy)] becomes zero. Clearly, [-⋁j≠iμϕNjP(xy)]=0 when ϕNi=Nr and xy is an Nr-edge. So(43)μNiϕPxy=μMPx∧μMPy,for xy∈Nr,ϕNi=Nr.From (41) and (43), the conclusion is obvious.
Definition 46.
Let Gˇb=(M,N1,N2,…,Nn) be a BFGS and let ϕ be any permutation on the set {1,2,…,n}. Then
Gˇb is self-complement if it is isomorphic to Gˇbϕc, the ϕ-complement of Gˇb;
Gˇb is strong self-complement if it is identical to Gˇbϕc.
Definition 47.
Let Gˇb=(M,N1,N2,…,Nn) be a BFGS. Then
Gˇb is totally self-complement if it is isomorphic to Gˇbϕc, the ϕ-complement of Gˇb, for all permutations ϕ on the set {1,2,…,n};
Gˇb is totally strong self-complement if it is identical to Gˇbϕc, the ϕ-complement of Gˇb, for all permutations ϕ on the set {1,2,…,n}.
Example 48.
All strong BFGSs are the only examples of self-complement or totally self-complement BFGSs.
Example 49.
A BFGS Gˇb=(M,N1,N2,N3) of graph structure G∗=(U,E1,E2,E3) as shown in Figure 10 is totally strong self-complement.
Totally strong self-complement BFGS.
Theorem 50.
A BFGS Gˇb is strong if and only if it is totally self-complement.
Proof.
Let Gˇb be a strong BFGS and ϕ any permutation on the set {1,2,…,n}.
By Theorem 45, Gˇbϕc is strong and if ϕ-1(i)=j, then all Ni-edges in Gˇb=(M,N1,N2,…,Nn) become Njϕ-edges in Gˇbϕc=M,N1ϕ,N2ϕ,…,Nnϕ(44)⟹μNiPa1a2=μMPa1∧μMPa2=μNjϕPa1a2,μNiNa1a2=μMNa1∨μMNa2=μNjϕNa1a2.Hence Gˇb is isomorphic to Gˇbϕc under the identity mapping f:U→U, such that μMP(a)=μMP(f(a)),μMN(a)=μMN(f(a))∀a∈U and(45)μNiPa1a2=μNjϕPa1a2=μNjϕPfa1fa2,μNiNa1a2=μNjϕNa1a2=μNjϕNfa1fa2,∀a1a2∈Ei,for ϕ-1i=j,i,j=1,2,…,n. This holds for any permutation on the set {1,2,…,n}.
Hence Gˇb is totally self-complement.
Conversely, let Gˇb and Gˇbϕc be isomorphic for any permutation ϕ on the set {1,2,…,n}. Then from the definition of ϕ-complement and isomorphism of BFGSs, we have(46)μNiPa1a2=μNjϕPfa1fa2=μMPfa1∧μMPfa2=μMPa1∧μMPa2,μNiNa1a2=μNjϕNfa1fa2=μMNfa1∨μMNfa2=μMNa1∨μMNa2∀a1a2∈Ei,i=1,2,…,n.
Hence, Gˇb is a strong BFGS.
Remark 51.
Every self-complement BFGS is necessarily totally self-complement.
Theorem 52.
If graph structure G∗=(U,E1,E2,…,En) is totally strong self-complement and M=(μMP,μMN) is a bipolar fuzzy set of U with constant valued functions μMP and μMN, then a strong BFGS Gˇb=(M,N1,N2,…,Nn) of G∗ is totally strong self-complement.
Proof.
Let s∈[0,1] and t∈[-1,0] be two constants, such that (47)μMPu=s,μMNu=t∀u∈U.Since G∗ is totally strong self-complement, so for every permutation ϕ-1 on the set {1,2,…,n}, there exists a bijection f:U→U, such that for every Ei-edge a1a2, “f(a1)f(a2)” [an Ej-edge in G∗] is an Ei-edge in (G∗)ϕ-1c and, consequently, for every Ni-edge a1a2, “f(a1)f(a2)” [a Nj-edge in Gˇb] is a Biϕ-edge in Gˇbϕc. Moreover Gˇb is strong, so we have(48)μMPa=s=μMPfa,μMNa=t=μMNfa∀u∈U,μNiPa1a2=μMPa1∧μMPa2=μMPfa1∧μMPfa2=μNjϕPfa1fa2,μNiNa1a2=μMNa1∨μMNa2=μMNfa1∨μMNfa2=μNjϕNfa1fa2∀a1a2∈Ei,i=1,2,…,n.
This shows that Gˇb is strong self-complement. This holds for any permutation ϕ and ϕ-1 on the set {1,2,…,n}; thus Gˇb is totally strong self-complement. This completes the proof.
Remark 53.
The converse of Theorem 52 is not necessary, since a totally strong self-complement BFGS Gˇb=(M,N1,N2,N3), as shown in Figure 10, is strong and has a totally strong self-complement underlying graph structure, but μMP and μMN are not constant valued functions.
4. Conclusions
Graph-theoretical concepts are widely used to study and model various applications in different areas. However, in many cases, some aspects of a graph-theoretical problem may be vague or uncertain. It is natural to deal with the vagueness and uncertainty using the methods of fuzzy sets. Since bipolar fuzzy set has shown advantages in handling vagueness and uncertainty than fuzzy set, we have applied the concept of bipolar fuzzy sets to graph structures. We have introduced the concept of bipolar fuzzy graph structures. We are extending our work to (1) bipolar fuzzy soft graph structures, (2) soft graph structures, (3) rough fuzzy soft graph structures, and (4) roughness in fuzzy graph structures.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors are thankful to the anonymous referees for the critical review of their paper.
SampathkumarE.Generalized graph structures20063265123MR2290946ZadehL. A.Fuzzy sets19658338353MR0219427ZhangW.-R.(Yin) (Yang) bipolar fuzzy sets1Proceedings of the IEEE International Conference on Fuzzy Systems Proceedings and the IEEE World Congress on Computational Intelligence (FUZZ-IEEE '98)May 1998Anchorage, Alaska, USAIEEE83584010.1109/FUZZY.1998.687599KauffmanA.19731Masson et CieRosenfeldA.ZadehL. A.FuK. S.ShimuraM.Fuzzy graphs1975New York, NY, USAAcademic Press7795BhattacharyaP.Some remarks on fuzzy graphs19876529730210.1016/0167-8655(87)90012-22-s2.0-0039795923MordesonJ. N.NairP. S.MordesonJ. N.19982ndHeidelberg, GermanyPhysicaSecond Edition 2001DineshT.2011Kannur, IndiaKannur UniversityAkramM.Bipolar fuzzy graphs2011181245548556410.1016/j.ins.2011.07.037MR28456712-s2.0-80054700052AkramM.Bipolar fuzzy graphs with applications2013391810.1016/j.knosys.2012.08.022AkramM.DudekW. A.Regular bipolar fuzzy graphs201221119720510.1007/s00521-011-0772-62-s2.0-84865790747AkramM.LiS.-G.ShumK. P.Antipodal bipolar fuzzy graphs20133197110MR31530292-s2.0-84892574672AkramM.DudekW. A.SarwarS.Properties of bipolar fuzzy hypergraphs201331426458MR31530322-s2.0-84892580020Al-ShehrieN. O.AkramM.Bipolar fuzzy competition graphs2015121385402LeeK.-M.Bipolar-valued fuzzy sets and their basic operationsProceedings of the International Conference2000Bangkok, Thailand307317MathewS.SunithaM. S.Node connectivity and arc connectivity of a fuzzy graph2010180451953110.1016/j.ins.2009.10.006MR25666092-s2.0-71149092663ZadehL. A.Similarity relations and fuzzy orderings197132177200MR0297650ZhangW.-R.Bipolar fuzzy sets and relations: a computational framework forcognitive modeling and multiagent decision analysisProceedings of the Industrial Fuzzy Control and Intelligent Systems Conference, and the NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, Fuzzy Information Processing Society Biannual ConferenceDecember 2014San Antonio, Tex, USA30530910.1109/IJCF.1994.375115