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FUZZY COMBINATORIAL DUAL GRAPH K. Shriram1, R. Sujatha1, S

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FUZZY COMBINATORIAL DUAL GRAPH K. Shriram1, R. Sujatha1, S International Journal of Pure and Applied Mathematics Volume 109 No. 7 2016, 35-41 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v109i7.5 ijpam.eu FUZZY COMBINATORIAL DUAL GRAPH K. Shriram1, R. Sujatha1, S. Narasimman1 1Department of Mathematics SSN College of Engineering Chennai, INDIA Abstract: Fuzzy graph is now a very important research area due to its wide application. Fuzzy planar graph is an important subclass of fuzzy graph theory. In this paper, we define the fuzzy combinatorial dual graph which is very close association with the fuzzy dual graph. The fuzzy combinatorial dual graph is defined and derived some of the results. AMS Subject Classification: 05C72, 03E72 Key Words: fuzzy graphs, fuzzy dual graphs, fuzzy combinatorial dual graphs 1. Introduction Graph theory is a way to demonstrate the modelling and Graph theory struc- tures are used to represent the telephone network, railway network, communica- tion problems, traffic network etc. A graph is a convenient way of representing the information involving relationship between objects. The objects are repre- sented by the set of vertices and the relationship between them is represented by the set of edges. When there is vagueness in the description of the objects or in its relationships or in both, it is natural that we need to construct a ”F uzzyGraphT heory”. In 1965, L.A. Zadeh [1] introduced a mathematical frame work to explain the concept of uncertainty in real life through the publication of a seminal paper ”F uzzysets”. A set of elements which are assigned with a degree of membership Received: October 1, 2016 c 2016 Academic Publications, Ltd. Published: February 17, 2016 url: www.acadpubl.eu 36 K. Shriram, R. Sujatha, S. Narasimman values are said to be fuzzy sets. A. Rosenfeld [2] introduced Fuzzy graph and also he coined many fuzzy analogous graph theoretic concepts like bridge, cut vertex and tree. Fuzzy graphs have many more applications in modelling real time systems where the level of information inherent in the system varies with different levels of precision. Mordeson and Nair [3] defined the concept of complement of fuzzy graphs and studied some operation on fuzzy graphs. Abdul-Jabbar et al. [4] introduced the concept of a fuzzy dual graph and discussed the some of the properties. Pal et al. [5] and Samanta et al. [6] introduced and investigated the concept of fuzzy planar graphs and studied several properties. Noura Alshehri et al. [7] introduced the concept of intuitionistic fuzzy planar graphs and discussed some of the properties. Frequent number of research works are done on this topic. 2. Basics Concepts and Preliminaries In this section, we see some elementary notions of graph theory and fuzzy graphs to understand the basic concepts mandatory for this paper. A graph [8] is a pair G = (V, E), where V is a non-empty set and E is a relation on V . A cut set of a graph is defined as the set of edges of a graph which, if removed , disconnects the graph. A multigraph is a graph that may contains multiple edges between any two vertices, but it does not contain any self loops. A drawing of a graph on any surface such that no edges intersect is called embedding. A graph G is planar if it can be drawn in the plane with its edges only inter- secting at vertices of G. A planar graph with cycles divides the plane into a set of regions called faces. The portion of the plane lying outside a graph embedded in a plane is infinite region. The dual graph of a plane graph G has a vertex corresponding to each face of G and an edge joining two neighbouring faces for each edge in G. The term dual is used because this property is symmetric, meaning that if H is a dual of G, then G is a dual of H(if G is connected). The duality concepts can be explained in different approaches. A dual graph can be classified into three types and they are geometrical dual graph, combinatorial dual graph and self-dual. A Combinatorial dual graph [8] of G is there is a one to one correspondence between their sets of edges which preserves cycles as sets of edges such that the removal of set of edges in G′ forms a cut set or bond. A fuzzy set [2] A drawn from X is defined as A = {< x, µA(x) >: x ∈ X}, where µ : X → [0, 1] is the membership function of the fuzzy set A. A fuzzy graph G = (V, σ, µ) is a nonempty set V together with a pair of functions σ : V → [0, 1] and µ : V × V → [0, 1] such that, for all x, y ∈ V ,µ(x, y) ≤ FUZZY COMBINATORIAL DUAL GRAPH 37 min(σ(x), σ(y)), where σ(x),σ(y) and µ(x, y) represents the membership values of the vertices and the edge in G, respectively. We denote the underlying graph by G∗ = (σ∗, µ∗) where σ∗ = {x ∈ V : σ(x) > 0} and µ∗ = {(x, y) ∈ V × V : µ(x, y) > 0}. A path P of length n is a sequence of distinct vertices x0, x1, . , xn such that µ(x(i−1), xi) > 0, i = 1, 2, . , n and the degree of the membership of a weakest edge is defined as its strength. The maximum of all strength of all paths between x and y is said to be connectedness of two vertices. If µ(w, x) = 1 and µ(y, z) = 0, then we say that the fuzzy graph has no crossing. Similarly, if µ(w, x) has value near to 1 and µ(y, z) has value near to 0, the crossing µ(y, z) will not be important for the planarity. If µ(y, z) has value near to 1 and µ(w, x) has value near to 1, then the crossing becomes very important for the planarity. Let V be a nonempty set and σ : V → [0, 1] is a mapping and E = n(x, y), µ(x, y)j, j = 1, 2, . , p|(x, y) ∈ V × V o be a fuzzy multiset [6] of V ×V such that µ(x, y)j ≤ min {(x), (y)} for all j = 1, 2, , p. Then G = (V, σ, E) is denoted as fuzzy multigraph [6] where σ(x) and µ(x, y)j represent the mem- bership value of the vertex x and the membership value of the edge (x, y) in G, respectively. Strength of the fuzzy edge I(x,y) can be measured by the value µ(x,y)j I(x,y) = min{σ(x),σ(y)} . Let G be a fuzzy multigraph and for a certain geometrical representation P1,P2,...,Pz be the points of the intersection between the edges. Then G is said to be fuzzy planar graph with fuzzy planarity value[6] f = 1 . 1+IP1 +IP2 ...+IPz 3. Fuzzy Dual Graphs The Fuzzy dual graph was introduced by Pal et al. [5] and Samanta et al. [6]. A different type of fuzzy planar graph was suggested by Samanta et al. [6] which is as called 0.67-fuzzy planar graph whose fuzzy planarity value is more than or equal to 0.67. When the fuzzy planarity value of a fuzzy graph is 1, then the geometrical representation of fuzzy planar graph is similar to the crisp planar graph or plane graph. By the theorem [6], if fuzzy planarity value is 0.67, then there is no crossing between the strong edges. For an instance, if there is any point of intersection between edges, that is the crossing between the weak edge and any other edge. The significance of the weak edge is less compared to strong edges. Thus, 0.67-fuzzy planar graph is more significant. If fuzzy planarity value increases, then the geometrical structure of planar graph tends to crisp planar graph. Any fuzzy planar graph without any point of 38 K. Shriram, R. Sujatha, S. Narasimman intersection of fuzzy edges is a fuzzy planar graph with fuzzy planarity value 1. Therefore, it is 0.67-fuzzy planar graph. Weak edges in 0.67-planar graph are not considered for any calculation in fuzzy dual graphs. Definition 1. [5] Let G = (V, σ, E) be a 0.67-fuzzy planar graph and E = {((x, y), (x, y)j), j = 1, 2, . , p|(x, y) ∈ V × V }, let F1,F2,...,Fk be the strong fuzzy faces of G. The fuzzy dual graph of G is ′ ′ ′ ′ ′ a fuzzy planar graph G = (V , σ ,E ) where V = {xi, i = 1, . , k}. The membership value of vertices are given by the mapping σ′ : V ′ → [0, 1] such ′ that σ (xi) = {max(u, v)j, j = 1, 2, p|(u, v)} is an edge of the boundary of strong fuzzy face Fi of G. Between two faces Fi and Fj of G, there may exist more than one common edge. Thus, between two vertices x and y in fuzzy dual graph G there may be more than one edge. The membership values of the fuzzy edges of the fuzzy dual graph are given by where (x, y)j = (u, v)j is an edge in the boundary between two strong fuzzy faces Fi and Fj and i = 1, 2, . , s where s is the number of common edges in the boundary between Fi and Fj or the number of common edges between x and y. 4. Fuzzy Combinatorial Dual Graphs In this section, Fuzzy combinatorial dual graph is defined and the related results are proved.
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