Intuitionistic Fuzzy Cycles and Intuitionistic Fuzzy Trees

Connectivity has an important role in neural networks, computer network, and clustering. In the design of a network, it is important to analyze connections by the levels. The structural properties of intuitionistic fuzzy graphs provide a tool that allows for the solution of operations research problems. In this paper, we introduce various types of intuitionistic fuzzy bridges, intuitionistic fuzzy cut vertices, intuitionistic fuzzy cycles, and intuitionistic fuzzy trees in intuitionistic fuzzy graphs and investigate some of their interesting properties. Most of these various types are defined in terms of levels. We also describe comparison of these types.


Introduction
A graph theory has many applications in different areas of computer science including data mining, image segmentation, clustering, image capturing, and networking. For example, a data structure can be designed in the form of trees; modeling of network topologies can be done using graph concepts. The most important concept of graph coloring is utilized in resource allocation and scheduling. The concepts of paths, walks, and circuits in graph theory are used in traveling salesman problem, database design concepts, and resource networking. This leads to the development of new algorithms and new theorems that can be used in tremendous applications.
A notion having certain influence on graph theory is fuzzy set, which is introduced by Zadeh [1] in 1965. Fuzzy graph theory is finding an increasing number of applications in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems.
Kaufmann's initial definition of a fuzzy graph [2] was based on Zadeh's fuzzy relations [3]. Rosenfeld [4] introduced the fuzzy analogue of several basic graph-theoretic concepts including bridges, cut nodes, connectedness, trees, and cycles. Bhattacharya [5] gave some remarks on fuzzy graphs, and Sunitha and Vijayakumar [6] characterized fuzzy trees. Bhutani and Rosenfeld [7] introduced the concepts of strong arcs, fuzzy end nodes, and geodesics in fuzzy graphs, and types of arcs in a fuzzy graph are described in [8]. Atanassov [9] introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs. Parvathi et al. [10,11] have studied intuitionistic fuzzy graphs and intuitionistic fuzzy shortest hyperpath in a network. Karunambigai et al. [12] have described arcs in intuitionistic fuzzy graphs. Akram et al. [13][14][15] have discussed many concepts, including strong intuitionistic fuzzy graphs, intuitionistic fuzzy hypergraphs, and metric aspects of intuitionistic fuzzy graphs. In this paper, we introduce various types of intuitionistic fuzzy bridges, intuitionistic fuzzy cut vertices, intuitionistic fuzzy cycles, and intuitionistic fuzzy trees in intuitionistic fuzzy graphs and investigate some of their interesting properties.

Preliminaries
In this section, we review some elementary concepts whose understanding is necessary to fully benefit from this paper. 2 The Scientific World Journal By a graph, we mean a pair * = ( , ), where is the set and is a relation on . The elements of are vertices of * and the elements of are edges of * . We write ∈ to mean that ( , ) ∈ , and if = ∈ , we say that and are adjacent. A path in a graph * is an alternating sequence of vertices and edges V 0 , 1 , V 1 , 2 , . . . , V −1 , , and V . The path graph with vertices is denoted by . A path is sometime denoted by Note that path graph, , has − 1 edges and can be obtained from cycle graph, , by removing any edge. An undirected graph * is connected if there is a path between each pair of distinct vertices. A block is a maximal biconnected subgraph of a given graph . An edge in a connected graph is a bridge (cut-edge or cut arc) if − is disconnected. A vertex V in a connected graph is a cut vertex if −V is disconnected. The graphs with exactly −1 bridges are exactly the trees, and the graphs in which every edge is a bridge are exactly the forests. A tree is a connected graph which contains no cycles.

Proposition 1.
Let be a graph with vertices. Then the following statements are equivalent.
(i) is connected and contains no cycles.
(ii) is connected and has − 1 edges.
(iii) has − 1 edges and contains no cycles.
(iv) is connected and each edge is a bridge.
(v) Any two vertices of are connected by exactly one path.
(vi) contains no cycles, but the addition of any new edge creates exactly one cycle.
A spanning tree in a connected graph is a subgraph of that includes all the vertices of and is also a tree. A forest is an undirected graph; all of its connected components are trees; in other words, the graph consists of a disjoint union of trees.
A fuzzy subset on a set is a map : ] ∞ ( , ) denotes the "strength of connectedness" between two nodes and . That is, ] ∞ ( , ) is defined as the maximum of the strengths of all paths between and .
In 1995, Atanassov [16] introduced the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets [1]. Atanassov added a new component (which determines the degree of nonmembership) in the definition of fuzzy set. The fuzzy sets give the degree of membership of an element in a given set (and the nonmembership degree equals one minus the degree of membership), while intuitionistic fuzzy sets give both a degree of membership and a degree of nonmembership which are more or less independent from each other; the only requirement is that the sum of these two degrees is not greater than 1.
When ( , ) = ] ( , ) = 0 for some , , there is no edge between and . Otherwise, there exists an edge between and .
Definition 7 (see [12]). An intuitionistic fuzzy graph is connected if any two vertices are joined by a path.

Bridges, Cut Vertices, and Blocks
Though the concept of path and connectedness in intuitionistic fuzzy graph is analogous to crisp graph, the other concepts like intuitionistic fuzzy tree and intuitionistic fuzzy bridge differ from those in crisp graph. In crisp graph, a cut node is the one whose removal from the graph disconnects the graph. A cut edge or bridge is also an edge whose removal disconnects the graph. But in intuitionistic fuzzy graph, the definitions of intuitionistic fuzzy bridge and intuitionistic fuzzy cut node are not so.
Definition 9 (see [12]). A bridge ( , ) in is said to be -bridge, if deleting ( , ) reduces the -strength of connectedness between some pair of vertices. A bridge ( , ) is said to be ]-bridge if deleting ( , ) increases the ]-strength of connectedness between some pair of vertices. A bridge ( , ) is said to be an intuitionistic fuzzy bridge if it is -bridge and ]-bridge.  Example 11. Consider a connected intuitionistic fuzzy graph as shown in Figure 1.
Routine computations show that connected intuitionistic fuzzy graph has no bridges of any of the five types.  We state the following propositions without their proofs.  Definition 21 (see [12]). A vertex ∈ in is called -cut vertex if deleting it reduces the -strength of connectedness between some pairs of vertices. A vertex ∈ is called ]-cut vertex if deleting it increases the ]-strength of connectedness between some pairs of vertices. A vertex ∈ is an intuitionistic fuzzy cut vertex if it is -cut vertex and ]-cut vertex.
The Scientific World Journal We state the following propositions without their proofs.

Proposition 26. Let be an intuitionistic fuzzy graph such that * is a cycle. Then a node is an intuitionistic fuzzy cut node of if and only if it is a common node of two intuitionistic fuzzy bridges.
Proposition 27. If is a common node of at least two intuitionistic fuzzy bridges, then is an intuitionistic fuzzy cut node.

Proposition 29. A complete intuitionistic fuzzy graph has no intuitionistic fuzzy cut vertex.
Definition 30. (1) is called a block if * is a block.
(2) is called an intuitionistic fuzzy block if it has no intuitionistic fuzzy cut vertices.
(3) is called a weak intuitionistic fuzzy block if there exists ( , ) ∈ (0, ℎ( )] such that ( , ) is a block.  Thus is a block and a weak intuitionistic fuzzy block. However, is not an intuitionistic fuzzy block since is an intuitionistic fuzzy cut vertex of . Also is not a partial intuitionistic fuzzy block since is a cut vertex for 0.5 < ≤ 0.9 and 0 < ≤ 0.1.
Example 35. All connected intuitionistic fuzzy graphs as shown in Figures 1, 2, 3, and 4 are firms.
Example 36. Consider a connected intuitionistic fuzzy graph as shown in Figure 5.  Thus is a block, an intuitionistic fuzzy block, and full intuitionistic fuzzy block. We note that is not firm. Example 38. Consider a connected intuitionistic fuzzy graph as shown in Figure 6. The proofs of the following propositions are trivial.  Thus is a partial intuitionistic fuzzy forest but is neither an intuitionistic fuzzy forest nor a full intuitionistic fuzzy forest.

Proposition 46. is a full intuitionistic fuzzy forest if and only if is forest.
Proof. Suppose that is a full intuitionistic fuzzy forest. Then * = ( ) is a forest. Conversely, suppose that is a forest. Then * is a forest and so must be ( , ) for all ( , ) ∈ (0, ℎ( )] since each ( , ) is a subgraph of * . This completes the proof. Proof. Suppose that contains a cycle whose edges are of strength ℎ( ). Then ( , ) , ( , ) ∈ (0, ℎ( )], that contains this cycle and so is not a forest. Thus is not a weak intuitionistic fuzzy forest.
Conversely, suppose that does not contain a cycle and all of its edges are of strength ℎ( ). Then ℎ( ) does not contain a cycle and so is a forest.

Corollary 49. If is an intuitionistic fuzzy forest, then is a weak intuitionistic fuzzy forest.
Theorem 50. is a forest and is constant on if and only if is a full intuitionistic fuzzy forest, * and ℎ( ) have the same number of connected components, and is firm.
Proof. Suppose that is a forest and is constant on . Then for all ( , ) ∈ (0, ℎ( )], ( , ) = * and so is a full intuitionistic fuzzy forest and * and ℎ( ) have the same number of connected components. Clearly, is firm since is a constant on .

Corollary 51. is a tree and is constant on if and only if
is a full intuitionistic fuzzy tree and is firm.  Thus is a partial intuitionistic fuzzy tree but not a full intuitionistic fuzzy tree.
is not an intuitionistic fuzzy tree.
We state the following propositions without their proofs.
Proposition 61. Suppose that is firm. If is a weak intuitionistic fuzzy tree, then is an intuitionistic fuzzy tree.
(2) is called intuitionistic fuzzy connected if is intuitionistic fuzzy block.

Proposition 63. If is connected, then is weakly connected.
Proof. connected implies that * is connected. Now * = ℎ( ) and so is weakly connected.
Proposition 64. If is firm and weakly connected, then is connected.
(2) is a tree if and only if is a forest and is connected.
(3) is partial intuitionistic fuzzy tree if and only if is a partial intuitionistic fuzzy forest and is partially intuitionistic fuzzy connected.

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(4) is a full intuitionistic fuzzy tree if and only if is a full intuitionistic fuzzy forest and is fully connected.
Proof. (1) If ( , ) is a tree for some ( , ) ∈ (0, ℎ( )], then ( , ) is connected and is a forest. For the converse, we note that ( 2 , 2 ) must also be a forest. Since also ( 2 , 2 ) is connected, ( 2 , 2 ) is a tree. The proofs of (2) Hence is firm.

Conclusions
In a network, each arc is assigned a weight. The weight of a path or a cycle is defined as the minimum weight of its arcs. The maximum of weights of all paths between two nodes is defined as the strength of connectedness between the nodes. In network applications, the reduction in the strength of connectedness is more relevant than the total disconnection of the graph. A graph is totally weighted if both node set and arc set are weighted. Fuzzy graph theory is finding an increasing number of applications in modeling real time systems. Since intuitionistic fuzzy models give more precision, flexibility, and compatibility to the system as compared to the fuzzy models, we have investigated some properties of intuitionistic fuzzy cycles, intuitionistic fuzzy trees, intuitionistic fuzzy bridges, and intuitionistic fuzzy cut vertices in intuitionistic fuzzy graphs in this paper. We plan to extend our research of fuzzification to (1) bipolar fuzzy trees, (2) soft cycles and soft trees, (3), and rough cycles and rough trees.