We apply the concept of m-polar fuzzy sets to graph structures. We introduce certain concepts in m-polar fuzzy graph structures, including strong m-polar fuzzy graph structure, m-polar fuzzy Di-cycle, m-polar fuzzy Di-tree, m-polar fuzzy Di-cut vertex, and m-polar fuzzy Di-bridge, and we illustrate these concepts by several examples. We present the notions of ϕ-complement of an m-polar fuzzy graph structure and self-complementary, strong self-complementary, totally strong self-complementary m-polar fuzzy graph structures, and we investigate some of their properties.
1. Introduction
Graph theory has 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 [2] is an important mathematical structure to represent a collection of objects whose boundary is vague. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional models used in engineering and science. Nowadays fuzzy sets are playing a substantial role in chemistry, economics, computer science, engineering, medicine, and decision-making problems. In 1997, Zhang [3] generalized the idea of a fuzzy set and gave the concept of bipolar fuzzy set on a given set X as a map which associates each element of X to a real number in the interval [-1,1]. In 2014, Chen et al. [4] introduced the idea of m-polar fuzzy sets as an extension of bipolar fuzzy sets and showed that bipolar fuzzy sets and 2-polar fuzzy sets are cryptomorphic mathematical notions and that we can obtain concisely one from the corresponding one in [4]. The idea behind this is that “multipolar information” (not just bipolar information which corresponds to two-valued logic) exists because data for a real world problem are sometimes from n agents (n≥2). For example, the exact degree of telecommunication safety of mankind is a point in [0,1]n(n≈7×109) because different person has been monitored different times. There are many examples such as truth degrees of a logic formula which are based on n logic implication operators (n≥2), similarity degrees of two logic formula which are based on n logic implication operators (n≥2), ordering results of a magazine, ordering results of a university, and inclusion degrees (accuracy measures, rough measures, approximation qualities, fuzziness measures, and decision preformation evaluations) of a rough set.
Kauffman [5] gave the definition of a fuzzy graph in 1973 on the basis of Zadeh’s fuzzy relations [6]. Rosenfeld [7] discussed the idea of fuzzy graph in 1975. Further remarks on fuzzy graphs were given by Bhattacharya [8]. Several concepts on fuzzy graphs were introduced by Mordeson and Nair [9]. In 2011, Akram introduced the notion of bipolar fuzzy graphs in [10]. Alshehri and Akram [11] introduced the concept of bipolar fuzzy competition graphs. In 2015, Akram and Younas studied certain types of irregular m-polar fuzzy graphs in [12]. Akram and Adeel studied m-polar fuzzy line graphs in [13]. Akram and Waseem introduced certain metrics in m-polar fuzzy graphs in [14]. Dinesh [15] introduced the notion of a fuzzy graph structure and discussed some related properties. Akram and Akmal [16] introduced the concept of bipolar fuzzy graph structures. In this paper, we apply the concept of m-polar fuzzy sets to graph structures. We introduce certain concepts in m-polar fuzzy graph structures, including strong m-polar fuzzy graph structure, m-polar fuzzy Di-cycle, m-polar fuzzy Di-tree, m-polar fuzzy Di-cut vertex, and m-polar fuzzy Di-bridge, and we illustrate these concepts by several examples. We present the notions of ϕ-complement of an m-polar fuzzy graph structure and self-complementary, strong self-complementary, totally strong self-complementary m-polar fuzzy graph structures, and we investigate some of their properties.
2. Preliminaries
In this section, we review some basic concepts that are necessary for fully benefit of this paper.
In 1965, Zadeh [2] introduced the notion of a fuzzy set as follows.
Definition 1 (see [<xref ref-type="bibr" rid="B16">2</xref>, <xref ref-type="bibr" rid="B17">6</xref>]).
A fuzzy set μ in a universe X is a mapping μ:X→[0,1]. A fuzzy relation on X is a fuzzy set ν in X×X. Let μ be a fuzzy set in X and ν fuzzy relation on X. We call ν a fuzzy relation on μ if ν(x,y)≤min{μ(x),μ(y)}∀x,y∈X.
Definition 2 (see [<xref ref-type="bibr" rid="B4">16</xref>]).
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 (1)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 sets of Gˇb, respectively.
Definition 3 (see [<xref ref-type="bibr" rid="B4">16</xref>]).
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},thatis,ϕ(Ni)=Nj, if and only if ϕ(Ei)=Ej∀i.
If xy∈Nr for some r and (2)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)andμNmϕN(xy)≤μNiϕN(xy)∀i.
And BFGS (M,N1ϕ,N2ϕ,…,Nnϕ), denoted by Gˇbϕc, is called the ϕ-complement of BFGS Gˇb.
Definition 4 (see [<xref ref-type="bibr" rid="B9">4</xref>]).
An m-polar fuzzy set (or a [0,1]m-set) on X is exactly a mapping A:X→[0,1]m.
Note that a [0,1]m-set is an L-set. An L-set on the set X is a synonym of a mapping A:X→L, where L is a lattice. So, [0,1]m is considered to be a partial order set with the point-wise order ≤, where m is an arbitrary ordinal number, ≤ is defined by x≤y⇔pi(x)≤pi(y) for each i∈m, and pi:[0,1]m→[0,1] is the ith projection mapping (i∈m). When L=[0,1], an L-set on X will be called a fuzzy set on X.
Definition 5 (see [<xref ref-type="bibr" rid="B5">14</xref>]).
Let C be an m-polar fuzzy subset of a nonempty set V. An m-polar fuzzy relation on C is an m-polar fuzzy subset D of V×V defined by the mapping D:V×V→[0,1]m such that, for all x,y∈V, pi∘D(xy)≤inf{pi∘C(x),pi∘C(y)},i=1,2,…,m, where pi∘C(x) denotes the ith degree of membership of the vertex x and pi∘D(xy) denotes the ith degree of membership of the edge xy.
Definition 6 (see [<xref ref-type="bibr" rid="B9">4</xref>, <xref ref-type="bibr" rid="B5">14</xref>]).
An m-polar fuzzy graph is a pair G=(C,D), where C:V→[0,1]m is an m-polar fuzzy set in V and D:V×V→[0,1]m is an m-polar fuzzy relation on V such that (3)pi∘Dxy≤infpi∘Cx,pi∘Cy for all x,y∈V.
We note that pi∘D(xy)=0 for all xy∈V×V-E for all i=1,2,3,…,m. C is called the m-polar fuzzy vertex set of G and D is called the m-polar fuzzy edge set of G, respectively. An m-polar fuzzy relation D on X is called symmetric if pi∘D(xy)=pi∘D(yx) for all x,y∈V.
3. Certain Concepts in <inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M156">
<mml:mrow>
<mml:mi>m</mml:mi></mml:mrow>
</mml:math></inline-formula>-Polar Fuzzy Graph StructuresDefinition 7.
Let G∗=(U,E1,E2,…,En) be a graph structure (GS). Let C be an m-polar fuzzy set on U and Di an m-polar fuzzy set on Ei such that(4)pj∘Dixy≤infpj∘Cx,pj∘Cyfor all x,y∈U, i∈n,j∈m, and pj∘Di(xy)=0 for xy∈U×U∖Ei,∀j. Then G(m)=(C,D1,D2,…,Dn) is called an m-polar fuzzy graph structure (m-PFGS) on G∗ where C is the m-polar fuzzy vertex set of G(m) and Di is the m-polar fuzzy i-edge set of G(m).
Definition 8.
Let G(m)=(C,D1,D2,…,Dn) and G(m′)=(C′,D1′,D2′,…,Dn′) be two m-PFGSs of a GS G∗=(U,E1,E2,…,En). Then G(m′) is called an m-polar fuzzy subgraph structure of G(m), if (5)pj∘C′x≤pj∘Cx,pj∘Di′xy≤pj∘Dixy,∀x,y∈U,j∈m,i∈n.G(m′) is called an m-polar fuzzy induced subgraph structure of G(m) by a set W⊆U, if(6)pj∘C′x=0,x∈U∖Wpj∘Cx,x∈W,∀j,pj∘Di′xy=0,x,y∈U∖Wpj∘Dixy,x,y∈W,∀j,∀i.G(m′) is called an m-polar fuzzy spanning subgraph structure of G(m), if C′=C and (7)pj∘Di′xy≤pj∘Dixyforj∈m,i∈n,x,y∈U.
Example 9.
Consider a graph structure G∗=(U,E1,E2) such that U={a1,a2,a3,a4}, E1={a1a2}, and E2={a3a2,a2a4}. Let C, D1, and D2 be 4-polar fuzzy sets on U, E1, and E2, respectively, defined by Tables 1 and 2.
By simple calculations, it is easy to check that G(m)=(C,D1,D2) is a 4-polar fuzzy graph structure of G∗ as shown in Figure 1. Note that we represent xy∈Di as (8)xyi=p1∘Dixy,…,pm∘Dixyiin all tables and the figures.
A 4-polar fuzzy induced subgraph structure of G(m) by U∖{a4} and a 4-polar fuzzy spanning subgraph structure of G(m) (Figure 1) are shown in Figure 2.
C
a1
a2
a3
a4
p1∘C
0.1
0.3
0.4
0.2
p2∘C
0.0
0.6
0.0
0.0
p3∘C
0.0
0.2
0.4
0.3
p4∘C
0.1
0.0
0.4
0.4
Di
(a1a2)1
(a3a2)2
(a2a4)2
p1∘Di
0.1
0.2
0.2
p2∘Di
0.0
0.0
0.0
p3∘Di
0.0
0.2
0.2
p4∘Di
0.0
0.0
0.0
Four-polar fuzzy graph structure.
Four-polar fuzzy subgraph structures.
Definition 10.
Let G(m)=(C,D1,D2,…,Dn) be an m-PFGS. Then xy∈Ei is called an m-polar fuzzy Di-edge or simply a Di-edge, if pj∘Di(xy)>0 for at least one j and we write that xy∈Di. Consequently, support of Di is defined by(9)suppDi=xy∈Ei:pj∘Dixy>0foratleastonej.An m-polar fuzzy Di-path, in m-PFGS G(m), is a sequence a1,a2,…,ak of vertices (distinct except the choice ak=a1) in U such that each aj-1aj is an m-polar fuzzy Di-edge for 1≤j≤k.
Definition 11.
A Di-strong m-PFGS is an m-polar fuzzy graph structure (10)Gm=C,D1,D2,…,Dn that satisfies(11)pj∘Dixy=infpj∘Cx,pj∘Cyfor all xy∈Di.G(m) is a strong m-polar fuzzy graph structure, if it is Di-strong for all i.
Definition 12.
An m-PFGS G(m)=(C,D1,D2,…,Dn) is called complete, if the following three conditions hold:
m-polar fuzzy graph structure G(m) is strong.
supp(Di)≠∅,∀i.
For each pair of vertices x,y∈U, xy is a Di-edge for some i.
Example 13.
Consider a graph structure G∗=(U,E1,E2) such that U={b1,b2,b3,b4}, E1={b1b2,b1b4,b3b4}, and E2={b3b2,b2b4,b1b3}. Let C, D1, and D2 be 4-polar fuzzy sets on U,E1, and E2, respectively, defined by Tables 3 and 4.
By simple calculations, it is easy to check that G(m)=(C,D1,D2) is a strong and complete 4-polar fuzzy graph structure of G∗, as shown in Figure 3. b2b1b4b3 is a 4-polar fuzzy D1-path consisting of all D1-edges constituting supp(D1)={b2b1,b1b4,b4b3}.
C
b1
b2
b3
b4
p1∘C
0.3
0.2
0.1
0.4
p2∘C
0.4
0.2
0.1
0.2
p3∘C
0.3
0.0
0.2
0.3
p4∘C
0.5
0.1
0.1
0.1
Di
(b1b2)1
(b1b4)1
(b3b4)1
(b1b3)2
(b2b4)2
(b3b2)2
p1∘Di
0.2
0.3
0.1
0.1
0.2
0.1
p2∘Di
0.2
0.2
0.1
0.1
0.2
0.1
p3∘Di
0.0
0.3
0.2
0.2
0.0
0.0
p4∘Di
0.1
0.1
0.1
0.1
0.1
0.1
A strong 4-polar fuzzy graph structure.
Definition 14.
Let G(m)=(C,D1,D2,…,Dn) be an m-PFGS and let j∈m. The j-strength or j-gain of a Di-path p=b1b2⋯bl, denoted by G·Pi,j, is defined by(12)G·PDi,j=⋀k=2lpj∘Dibk-1bk.
Example 15.
Consider the 4-PFGS G(m)=(C,D1,D2), shown in Figure 3, in which P=b2b1b4b3 is a 4-polar fuzzy D1-path. Then j-gain, for j=1,2,3,4, of this path P is given by G·PD1,1=0.1, G·PD1,2=0.1, G·PD1,3=0.0, and G·PD1,4=0.1.
Definition 16.
Let G(m)=(C,D1,D2,…,Dn) be an m-polar fuzzy graph structure and let j∈m. Then Di,j-strength of connectedness between two vertices x and y is defined by pj,i∞(xy)=⋁k≥1{pj,ik(xy)}, where for k≥2 and pj,i1(xy)=pj∘Di(xy)(13)pj,ikxy=pj,ik-1∘pj,i1xy=⋁zpj,ik-1xz∧pj,i1zy, for k≥2 and pj,i1(xy)=pj∘Di(xy).
Example 17.
Consider the 4-PFGS G(m)=(C,D1,D2) as shown in Figure 3. All the calculated Di,j-strengths of connectedness between any two vertices of G(m), for i=1,j=1, are given in Table 5.
Connectedness between vertices.
(b1b2)1
(b1b3)2
(b1b4)1
(b2b4)2
(b2b3)2
(b3b4)1
p1,11
0.2
0.0
0.3
0.0
0.0
0.1
p1,12
0.2
0.1
0.0
0.2
0.0
0.0
p1,13
0.0
0.0
0.1
0.2
0.1
0.1
p1,14
0.1
0.1
0.0
0.1
0.1
0.0
p1,15
0.0
0.0
0.1
0.1
0.1
0.1
p1,16
0.1
0.1
0.0
0.1
0.1
0.0
p1,1∞
0.2
0.1
0.3
0.2
0.1
0.1
Definition 18.
An m-polar fuzzy Di-cycle is an m-PFGS G(m)=(C,D1,D2,…,Dn) such that
(supp(C),supp(D1),supp(D2),…,supp(Dn)) is an Ei-cycle (it is called a Di-cycle),
There is no unique Di-edge uv in G(m) such that for any j∈m(14)pj∘Diuv=minpj∘Dixy:xy∈suppDi.
Example 19.
Let G(m)=(C,D1,D2) be the 4-PFGS, shown in Figure 4; then G(m) is a 4-polar fuzzy D2-cycle.
A 4-polar fuzzy D2-cycle.
Definition 20.
Let G(m)=(C,D1,D2,…,Dn) be an m-polar fuzzy graph structure and let x be a vertex in G(m). Let (C′,D1′,D2′,…,Dn′) be an m-polar fuzzy subgraph structure of G(m) induced by U∖{x}, such that for i∈n,j∈m(15)pj∘C′x=0,pj∘Di′xv=0,foralledgesxvinGm,(16)pj∘C′v=pj∘Cv,pj∘Di′uv=pj∘Diuv,forv≠x,u≠x. Then x is called an m-polar fuzzy Di,j-cut vertex for some i, if for some u,v∈U∖{x}(17)pj,i∞uv>pj,i∞′uvand x is called an m-polar fuzzy Di-cut vertex for some i, if it is Di,j-cut vertex for all j.
Definition 21.
Let G(m)=(C,D1,D2,…,Dn) be an m-polar fuzzy graph structure and xy an edge in G(m). Let (C,D1′,D2′,…,Dn′) be an m-polar fuzzy spanning subgraph structure of G(m) such that(18)pj∘Di′xy=0,pj∘Di′uv=pj∘Diuv,foredgesuv≠xy,i∈n,j∈m.Then xy is an m-polar fuzzy Di,j-bridge for some i, if for some u,v∈U(19)pj,i∞uv>pj,i∞′uv.And xy is called an m-polar fuzzy Di-bridge for some i, if it is Di,j-bridge for all j.
Example 22.
Consider the 4-PFGS G(m)=(C,D1,D2) as shown in Figure 3. The vertex b1 is a 4-polar fuzzy D1,1-cut vertex since p1,1∞(b2b3)=0.1>0.0=p1,1∞′(b2b3) and p1,1∞(b2b4)=0.2>0.0=p1,1∞′(b2b4), where (W,D1′,D2′) is the 4-polar fuzzy subgraph structure of G(m) induced by W=C∖{b1}.
The edge b1b4 is a 4-polar fuzzy D1,1-bridge since p1,1∞(b1b4)=0.3>0.0=p1,1∞′(b1b4), p1,1∞(b1b3)=0.1>0.0=p1,1∞′(b1b3), p1,1∞(b2b3)=0.1>0.0=p1,1∞′(b2b3), and p1,1∞(b2b4)=0.2>0.0=p1,1∞′(b2b4), where (C,D1′,D2′) is the 4-polar fuzzy spanning subgraph structure of G(m) that excludes the D1-edge b1b4.
Definition 23.
An m-PFGS G(m)=(C,D1,D2,…,Dn) is a Di-tree if (supp(C),supp(D1),supp(D2),…,supp(Dn)) is an Ei-tree.
Definition 24.
An m-PFGS G(m)=(C,D1,D2,…,Dn) is an m-polar fuzzy Di-tree if there is an m-polar fuzzy spanning subgraph structure G(m)′=(C′,D1′,D2′,…,Dn′) such that G(m)′ is a Di′-tree and for every edge xy not belonging to G(m)′(20)pj∘Dixy<pj,i∞′xy∀j. If the above condition holds for j∈m, G(m) is called a Di,j-tree.
Example 25.
Consider the 4-PFGS G(m)=(C,D1,D2) as shown in Figure 5.
Then G(m) is a 4-polar fuzzy D1,1-tree since p1∘D1(b2b5)=0.2<0.3=p1,1∞′(b2b5), where (C,D1′,D2′) is the 4-polar fuzzy spanning subgraph structure of G(m) that excludes the D1-edge b2b5. Also G(m) is a D2-tree.
A 4-polar fuzzy D1,1-tree.
Definition 26.
An m-PFGS G(m)1=(C1,D11,D12,…,D1n) of a graph structure G∗1=(U1,E11,E12,…,E1n) is said to be isomorphic to an m-PFGS G(m)2=(C2,D21,D22,…,D2n) of a graph structure G∗2=(U2,E21,E22,…,E2n), if there exists a bijection h:U1→U2 and a permutation ϕ on the set {1,2,…,n} such that for k∈m,i∈n(21)pk∘C1u1=pk∘C2hu1,∀u1∈U1,pk∘D1iu1u2=pk∘D2ϕihu1hu2,∀u1u2∈E1i.
Example 27.
Two isomorphic 4-PFGSs G(m1)=(C1,D11,D12) and G(m2)=(C2,D21,D22) are shown in Figure 6. This isomorphism holds under the permutation ϕ=(12) and the mapping h:U1→U2, defined by h(a1)=b3,h(a2)=b4,h(a3)=b1, and h(a4)=b2.
Two isomorphic 4-polar fuzzy graph structures.
Definition 28.
An m-PFGS G(m)1=(C1,D11,D12,…,D1n) is said to be identical to an m-PFGS G(m)2=(C2,D21,D22,…,D2n) if there exists a bijection h:U→U, such that for j∈m,i∈n(22)pj∘C1u=pj∘C2hu,∀u∈U,pj∘D1iu1u2=pj∘D2ihu1hu2,∀u1u2∈E1i.
Example 29.
Two 4-PFGSs G(m1)=(C1,D11,D12) and G(m2)=(C2,D21,D22), identical under the mapping h:U1→U2, defined by h(a1)=b3,h(a2)=b4,h(a3)=b1, and h(a4)=b2, are shown in Figure 7.
Identical 4-polar fuzzy graph structures.
Definition 30.
Let G(m)=(C,D1,D2,…,Dn) be an m-polar fuzzy graph structure. Let ϕ be any permutation on the set {1,2,…,n} and ϕ(Di)=Dj⇔ϕ(i)=j. If xy∈Dr for some r∈n and (23)pj∘Diϕxy=infpj∘Cx,pj∘Cy-⋁k≠ipj∘Dϕkxy for j∈m,i∈n then xy∈Dsϕ, where s is chosen such that, for each j, (24)pj∘Dsϕxy=maxpj∘Diϕxy:i∈n.Then an m-PFGS constructed with C and nm-polar fuzzy relations (Dsϕ,s=1,2,…,n) on C, denoted by G(m)ϕc=(C,D1ϕ,D2ϕ,…,Dnϕ), is called the ϕ-complement of m-polar fuzzy graph structure G(m).
Proposition 31.
A ϕ-complement of an m-polar fuzzy graph structure is always a strong m-PFGS. Moreover, if ϕ-1(r)=i for r,i∈n, then all Dr-edges in m-PFGS G(m)=(C,D1,D2,…,Dn) become Diϕ-edges in G(m)ϕc=(C,D1ϕ,D2ϕ,…,Dnϕ).
Proof.
From the definition of ϕ-complement G(m)ϕc of an m-PFGS G(m), (25)pj∘Diϕxy=infpj∘Cx,pj∘Cy-⋁k≠ipj∘Dϕkxy,j∈m,i∈n.We can see that inf(pj∘C(x),pj∘C(y))≥0,⋁k≠ipj∘Dϕ(k)(xy)≥0, and pj∘Dk(xy)≤inf(pj∘C(x),pj∘C(y)),∀Dk. So(26)⋁k≠ipj∘Dϕkxy≤infpj∘Cx,pj∘Cy⟹pj∘Diϕxy=infpj∘Cx,pj∘Cy-⋁k≠ipj∘Dϕkxy≥0∀i,j.Moreover, the maximum value of pj∘Diϕ(xy) occurs when its negative part (-⋁k≠ipj∘Dϕ(k)(xy)) becomes zero, and it is zero when ϕ(i)=r and xy is a Dr-edge in G(m). Therefore, pj∘Diϕ(xy)=inf(pj∘C(x),pj∘C(y)) for xy∈Dr,ϕ(i)=r.
Hence G(m)ϕc=(C,D1ϕ,D2ϕ,…,Dnϕ) is a strong m-PFGS and every Dr-edge in G(m) becomes Diϕ-edge for ϕ-1(r)=i.
Remark 32.
If G(m) is an m-polar fuzzy graph structure of a GS G∗ then ϕ-complement of G(m) is a strong m-polar fuzzy graph structure of ϕ-1-complement of G∗.
Definition 33.
Let G(m)=(C,D1,D2,…,Dn) be an m-polar fuzzy graph structure and ϕ be any permutation on the set {1,2,…,n}. Then
G(m) is called self-complementary if G(m) is isomorphic to G(m)ϕc,
G(m) is called strong self-complementary if G(m) is identical to G(m)ϕc.
Definition 34.
Let G(m)=(C,D1,D2,…,Dn) be an m-polar fuzzy graph structure. Then
G(m) is called totally self-complementary if G(m) is isomorphic to G(m)ϕc for every permutation ϕ on the set 1,2,…,n,
G(m) is called totally strong self-complementary if G(m) is identical to G(m)ϕc for every permutation ϕ on the set {1,2,…,n}.
Theorem 35.
An m-PFGS G(m) is strong if and only if it is totally self-complementary.
Proof.
We suppose that G(m) is a strong m-PFGS of a GS G∗=(U,E1,E2,…,En) and ϕ is any permutation on the set {1,2,…,n}.
By Proposition 31, G(m)ϕc is strong and all Di-edges in G(m) become Djϕ-edges for ϕ-1(i)=j. So (27)pk∘Dia1a2=infpk∘Ca1,pk∘Ca2=pk∘Djϕa1a2,∀k. Using the identity mapping h:U→U, we get pk∘C(a)=pk∘C(h(a)),∀k∈m,a∈U, and (28)pk∘Dia1a2=infpk∘Ca1,pk∘Ca2=infpk∘Cha1,pk∘Cha2=pk∘Djϕha1ha2,∀k.That is,(29)pk∘Dia1a2=pk∘Djϕha1ha2,∀k.Hence G(m) is isomorphic to G(m)ϕc under the identity mapping h:U→U and a permutation ϕ. Since ϕ was an arbitrary permutation on {1,2,…,n}, G(m) is totally self-complementary.
Conversely suppose that G(m) is a totally self-complementary m-PFGS and we have to prove that G(m) is strong m-PFGS.
Since G(m) and G(m)ϕc are isomorphic for all permutations on {1,2,…,n}, therefore by definition of isomorphism pk∘Di(a1a2)=pk∘Djϕ(h(a1)h(a2))=inf(pk∘C(h(a1)),pk∘C(h(a2)))=inf(pk∘C(a1),pk∘C(a2)), for all a1a2∈Ei,i∈n,k∈m. Hence G(m) is a strong m-PFGS.
This completes the proof.
Theorem 36.
Let G(m) be a strong m-PFGS on a GS G∗=(U1,E11,E12,…,E1n). If G∗ is totally strong self-complementary and C is an m-polar fuzzy set on U such that for each j∈m,pj∘C assigns a constant value cj∈[0,1] to every u∈U, then G(m) is totally strong self-complementary.
Proof.
We assume that G∗ is totally strong self-complementary and pj∘C(u)=cj for all u∈U, where cj∈[0,1] is a constant for each j∈m.
Since G∗ is totally strong self-complementary, for any permutation ϕ-1 with ϕ-1(k)=i, there exists a bijection h:U→U, such that, for every Ei-edge a1a2 in G∗, h(a1)h(a2) (an Ek-edge in G∗) is an Ei-edge in (G∗)ϕ-1c. Consequently, for every Di-edge a1a2 in G(m), h(a1)h(a2) (a Dk-edge in G(m)) is a Diϕ-edge in G(m)ϕc. Moreover, (30)pj∘Cha=cj=pj∘Ca,∀a,ha∈U,j∈m and G(m) is a strong m-PFGS, so we get (31)pj∘Dia1a2=infpj∘Ca1,pj∘Ca2=infpj∘Cha1,pj∘Cha2=pj∘Diϕha1ha2,∀a1a2∈Ei,i∈n,j∈m.This shows that G(m) is identical to G(m)ϕc. Hence G(m) is totally strong self-complementary, since ϕ was an arbitrary permutation. This completes the proof.
Converse of Theorem 36 does not hold because every strong and totally strong self-complementary m-PFGS does not necessarily include an m-polar fuzzy vertex set C such that for each j∈m,pj∘C assigns a constant value cj∈[0,1] to every u∈U. This can be observed in m-PFGS shown in Figure 9.
Example 37.
There are no other totally self-complementary m-PFGSs than strong m-polar fuzzy graph structures. So all strong m-PFGSs are the examples of totally self-complementary m-PFGSs.
Example 38.
An m-PFGS, shown in Figure 8, is strong self-complementary but not totally strong self-complementary because this m-PFGS is identical to its ϕ-complement only when ϕ=(12)(34).
An m-PFGS, shown in Figure 9, is totally strong self-complementary.
A strong self-complementary 4-PFGS.
A totally strong self-complementary 4-PFGS.
4. Conclusions
A graph structure is a useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. It helps to study various relations and the corresponding edges simultaneously. Sometimes information in a network model is based on multiagent, multiattribute, multiobject, multipolar information or uncertainty rather than a single bit. An m-polar fuzzy model is useful for such network models which gives more and more precision, flexibility, and comparability to the system as compared to the classical, fuzzy, and bipolar fuzzy models. We have introduced certain concepts in m-polar fuzzy graph structures. We are extending our work to (1) neutrosophic graph structures, (2) intuitionistic neutrosophic soft graph structures, (3) roughness in graph structures, and (4) intuitionistic fuzzy soft graph structures.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this article.
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