Abstract
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.
1. 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. [13–15] 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.
We have used standard definitions and terminologies in this paper. For other notations, terminologies, and applications not mentioned in the paper, the readers should refer to [3, 5, 7, 8, 10, 16–23].
2. Preliminaries
In this section, we review some elementary concepts whose understanding is necessary to fully benefit from this paper.
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 , and . The path graph with vertices is denoted by . A path is sometime denoted by . The length of a path in is . A path in is called a cycle if and . Note that path graph, , has 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 in a connected graph is a cut vertex if is disconnected. The graphs with exactly 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 edges.(iii) has 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 . A fuzzy binary relation on is a fuzzy subset on . By a fuzzy relation , we mean a fuzzy binary relation given by . Let be a fuzzy set of defined by . Then is called the composition of with itself. Since composition is associative, we get for . Define the fuzzy subset of by 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.
An intuitionistic fuzzy set (IFS, for short) on a universe is an object of the form where is called degree of membership of in , is called degree of nonmembership of in , and , satisfies the following condition for all , . In particular, we use and to denote the intuitionistic fuzzy empty set and the intuitionistic fuzzy whole set in a set such that and , for each , respectively. An intuitionistic fuzzy relation in a universe (, for short) is an intuitionistic fuzzy set of the form where and . The intuitionistic fuzzy relation satisfies for all , . An intuitionistic fuzzy relation on universe is called reflexive if = (1, 0) for each . is called symmetric if for any , .
Definition 2. The height of an intuitionistic fuzzy set is defined as
We will say that intuitionistic fuzzy set is normal if there is at least one such that = 1. The depth of an intuitionistic fuzzy set is defined as
Notation. (1) Let ; then means that .
(2) means that .
Definition 3 (see [14]). By an intuitionistic fuzzy graph (IFG), one means a pair in which is an intuitionistic fuzzy set on and is an intuitionistic fuzzy relation on such that , , and for all . Note that is a symmetric intuitionistic fuzzy relation on .
Definition 4 (see [12, 14]). An intuitionistic fuzzy graph is called complete if and for all , .
Definition 5 (see [9]). The support of is defined by The support of is defined by Let . For , is called an -level subset of and is called an -level subset of . Let .
Definition 6 (see [12]). A path in a intuitionistic fuzzy graph is an sequence of distinct vertices such that either one of the following condition is satisfied:(1) and for some , ;(2) and for some , ;(3) and for some , .
When 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.
Definition 8 (see [12]). If , the -strength of connectedness between and is The -strength of connectedness between and is The -strength and -strength of connectedness between and in are denoted by and , respectively. Also and denote and , where is obtained from by deleting the arc .
3. 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.
Definition 10. Let . (1) is called a bridge if is a bridge of .(2) is called an intuitionistic fuzzy bridge if and for some , where and are and restricted to .(3) is called a weak intuitionistic fuzzy bridge if there exists such that is a bridge of .(4) is called a partial intuitionistic fuzzy bridge if is a bridge for for all .(5) is called a full intuitionistic fuzzy bridge if is a bridge for for all .
Example 11. Consider a connected intuitionistic fuzzy graph as shown in Figure 1.
By routine computations, we have and . Thus means that . For , , . For , , . Hence we conclude that is a full intuitionistic fuzzy bridge and is a weak intuitionistic fuzzy bridge but not a partial intuitionistic fuzzy bridge. Both and are bridges and intuitionistic fuzzy bridges.
Example 12. Consider a connected intuitionistic fuzzy graph as shown in Figure 2.
By routine computations, we have and . For , , . For , , . For , , . Thus is an intuitionistic fuzzy bridge and a partial intuitionistic fuzzy bridge but not a bridge. The edge is not any of five types of bridges.
Example 13. Consider a connected graph such that and . Let be an intuitionistic fuzzy set of and let be an intuitionistic fuzzy set of defined by Routine computations show that connected intuitionistic fuzzy graph has no bridges of any of the five types.
Example 14. Consider a connected graph such that and . Let be an intuitionistic fuzzy set of and let be an intuitionistic fuzzy set of defined by By routine computations, we have and . For , , . For , , . Thus is a full intuitionistic fuzzy bridge and is a partial intuitionistic fuzzy bridge but not a full intuitionistic fuzzy bridge.
We state the following propositions without their proofs.
Proposition 15. Let be a bridge in . Then is an intuitionistic fuzzy bridge if and only if and .
Proposition 16. is an intuitionistic fuzzy bridge if and only if is not the weakest bridge of any cycle.
Proposition 17. is an intuitionistic fuzzy bridge if and only if is a bridge for and and .
Proof. Suppose that is a full bridge. Then is a bridge for for all . Hence and so and . Since is a bridge for for all , it follows that is a bridge for since and .
Conversely, suppose that is a bridge for and and . Then for all . Thus since also is a bridge for , is a bridge for for all since each is a subgraph of . Hence is a full intuitionistic fuzzy bridge.
Proposition 18. Suppose that is not contained in a cycle of . Then the following conditions are equivalent: (1) and ;(2) is a partial intuitionistic fuzzy bridge;(3) is an intuitionistic full fuzzy bridge.
Proof. Since is not contained in a cycle of , is a bridge of . Hence by Proposition 17, (1) (3). Clearly, (3) (2). Suppose that (2) holds. Then is a bridge for for all and so . Thus and ; that is, (1) holds.
Proposition 19. If is a bridge, then is a weak intuitionistic fuzzy bridge and an intuitionistic fuzzy bridge.
Proposition 20. is an intuitionistic fuzzy bridge if and only if is a weak bridge.
Proof. Suppose that is a weak intuitionistic fuzzy bridge. Then such that is a bridge for . Hence removal of disconnects . Thus any path from to in has an edge with , . Thus the removal of results in , . Hence is an intuitionistic fuzzy bridge.
Conversely, suppose that is an intuitionistic fuzzy bridge. Then such that removal of results in , . Hence is on every strongest path connecting and and in fact and this value. Thus there does not exist a path (other than ) connecting and in , else this other path without would be of strength and and would be part of a path connecting and of strongest length, contrary to the fact that is on every such path. Hence is a bridge of and , . Thus and are desired .
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.
Definition 22. Let .(1) is called a cut vertex if is a cut-vertex of .(2) is called an intuitionistic fuzzy cut-vertex if and for some , where and are and restricted to .(3) is called a weak intuitionistic fuzzy cut-vertex if there exists such that is a cut-vertex of .(4) is called a partial intuitionistic fuzzy cut-vertex if is a cut-vertex for for all .(5) is called a full intuitionistic fuzzy cut-vertex if is a cut-vertex for for all .
Example 23. Consider a connected intuitionistic fuzzy graph as shown in Figure 3.
By routine computations, we have , . Thus means that . For and , . For and , . For and , . Thus is an intuitionistic fuzzy cut-vertex and a weak intuitionistic fuzzy cut-vertex but neither a cut-vertex nor a partial cut-vertex.
Example 24. Consider a connected graph such that and . Let be an intuitionistic fuzzy set of and let be an intuitionistic fuzzy set of defined by By routine computations, we have and . For and , . For and , . Thus is an intuitionistic fuzzy cut-vertex and a partial intuitionistic fuzzy cut-vertex but neither a cut-vertex nor a full cut-vertex.
Example 25. Consider a connected graph such that and . Let be an intuitionistic fuzzy set of and let be an intuitionistic fuzzy set of defined by By routine computations, we have and . For and , . Thus is a full intuitionistic fuzzy cut-vertex, an intuitionistic fuzzy cut-vertex, and a cut-vertex.
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 28. If is a complete intuitionistic fuzzy graph, then and .
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 such that is a block.
(4) is called a partial intuitionistic fuzzy block if is a block for all .
(5) is called a full intuitionistic fuzzy block if is a block for all .
Example 31. Consider a connected intuitionistic fuzzy graph as shown in Figure 4.
By routine computations, we have and . Thus means that . For and , . For and , . Thus is a block, an intuitionistic fuzzy block, and a weak intuitionistic fuzzy block. is not a partial intuitionistic fuzzy block since is not a block for , .
Example 32. Consider a connected graph such that and . Let be an intuitionistic fuzzy set of and let be an intuitionistic fuzzy set of defined by By routine computations, we have and . For and , . For and , . 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 and .
Example 33. Consider a connected graph such that and . Let be an intuitionistic fuzzy set of and let be an intuitionistic fuzzy set of defined by By routine computations, we have and . For and , . Thus is a block, an intuitionistic fuzzy block, and a full intuitionistic fuzzy block.
Definition 34. A connected intuitionistic fuzzy graph is said to be firm if
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.
By routine computations, we have and . Thus means that . For and , . For and , . Thus is a block, an intuitionistic fuzzy block, and full intuitionistic fuzzy block. We note that is not firm.
4. Cycles and Trees
Definition 37. (1) is called a cycle if is a cycle.
(2) is called an intuitionistic fuzzy cycle if is a cycle and there does not exist unique such that , .
(3) is called a weak intuitionistic fuzzy cycle if there exists such that is a cycle.
(4) is called a partial intuitionistic fuzzy cycle if is a cycle for all .
(5) is called a full intuitionistic fuzzy cycle if is a cycle for all .
Example 38. Consider a connected intuitionistic fuzzy graph as shown in Figure 6.
By routine computations, we have and . Thus means that . For and , . For and , . Thus is an intuitionistic fuzzy cycle and weak intuitionistic fuzzy cycle but is not partial intuitionistic fuzzy cycle.
Example 39. Consider a connected intuitionistic fuzzy graph as shown in Figure 7.
By routine computations, we have and . Thus means that . For and , which is not a cycle. For and , which is a cycle. Thus is not cycle; is a partial intuitionistic fuzzy cycle but not a full intuitionistic fuzzy cycle.
The proofs of the following propositions are trivial.
Proposition 40. Suppose that is a cycle. Then is a partial intuitionistic fuzzy cycle if and only if is a full intuitionistic fuzzy cycle.
Proposition 41. is a full intuitionistic fuzzy cycle if and only if is a cycle and is constant on .
Proposition 42. is a partial intuitionistic fuzzy cycle if and only if is a cycle and .
Proof. Suppose that is a partial intuitionistic fuzzy cycle. Then clearly is a cycle and in fact is a cycle for all . Suppose that . Then such that and . Hence such that , . Thus and so is not a cycle, a contradiction.
Conversely, suppose that is a cycle and . If , then is a full intuitionistic fuzzy cycle by Proposition 41. Suppose that . Then . Since for and , it follows that is a partial intuitionistic fuzzy cycle.
Definition 43. A connected intuitionistic fuzzy graph is an intuitionistic fuzzy tree if it has an intuitionistic fuzzy spanning subgraph which is a tree, where for all arcs not in , , .
Definition 44. (1) is called a forest if is a forest.
(2) is called an intuitionistic fuzzy forest if has an intuitionistic fuzzy spanning subgraph which is a forest such that, for all , and .
(3) is called a weak intuitionistic fuzzy forest if for all such that is a forest.
(4) is called a partial intuitionistic fuzzy forest if is a forest for all .
(5) is called a full intuitionistic fuzzy forest if is a forest for all .
Example 45. Consider a connected graph such that and . Let be an intuitionistic fuzzy set of and let be an intuitionistic fuzzy set of defined by By routine computations, we have and . For and , , and for and , . 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 since each is a subgraph of . This completes the proof.
Example 47. Consider a connected graph such that and . Let be an intuitionistic fuzzy set of and let be an intuitionistic fuzzy set of defined by By routine computations, we have and . For and , . For and , . Thus is a forest and a full intuitionistic fuzzy forest without being a constant on . Note that has more connected components than .
Proposition 48. is a weak intuitionistic fuzzy forest if and only if does not contain a cycle whose edges are of strength .
Proof. Suppose that contains a cycle whose edges are of strength . Then , , 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 . Then 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 have the same number of connected components, and is firm.
Proof. Suppose that is a forest and is constant on . Then for all , and so is a full intuitionistic fuzzy forest and and have the same number of connected components. Clearly, is firm since is a constant on .
Conversely, suppose that is a full intuitionistic fuzzy forest, and have the same number of connected components, and is firm. Suppose that , such that . Then such that and . Now , . Hence has more connected components then since is firm; that is, no vertices were lost. Thus has more connected components than , a contradiction.
Corollary 51. is a tree and is constant on if and only if is a full intuitionistic fuzzy tree and is firm.
Definition 52. (1) is called a tree if is a tree.
(2) is called an intuitionistic fuzzy tree if has an intuitionistic fuzzy spanning subgraph which is a tree such that, for all , and .
(3) is called a weak intuitionistic fuzzy tree if for all such that is a tree.
(4) is called a partial intuitionistic fuzzy tree if is a tree for all .
(5) is called a full intuitionistic fuzzy tree if is a tree for all .
Example 53. Consider a connected graph such that and . Let be an intuitionistic fuzzy set of and let be an intuitionistic fuzzy set of defined by By routine computations, we have and . For and , , and for and , . Thus is a tree, is a full intuitionistic fuzzy tree, and and have the same number of connected components. However, is not firm and is not constant on .
Example 54. Consider a connected graph such that and . Let be an intuitionistic fuzzy set of and let be an intuitionistic fuzzy set of defined by By routine computations, we have and . For and , , and for and , . 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 55. If is an intuitionistic fuzzy tree, then G is not complete.
Proposition 56. If is an intuitionistic fuzzy tree, then arcs of spanning subgraph are the intuitionistic fuzzy bridges of .
Proposition 57. If is an intuitionistic fuzzy tree, then internal nodes of spanning subgraph are the intuitionistic fuzzy cut nodes of .
Proposition 58. is an intuitionistic fuzzy tree if and only if the following are equivalent: (a) is an intuitionistic fuzzy bridge;(b) and .
Proposition 59. An intuitionistic fuzzy graph is an intuitionistic fuzzy tree if and only if it has a unique maximum spanning tree.
Definition 60. For all , one defines and by
Proposition 61. Suppose that is firm. If is a weak intuitionistic fuzzy tree, then is an intuitionistic fuzzy tree.
Proof. There exist such that is a tree. Since is firm, is an intuitionistic fuzzy spanning subgraph of which is a tree. If is in , then , and so it follows that is an intuitionistic fuzzy tree.
Definition 62. (1) is called connected if is connected.
(2) is called intuitionistic fuzzy connected if is intuitionistic fuzzy block.
(3) is called weak intuitionistic fuzzy connected if there exists such that is connected.
(4) is called partial intuitionistic fuzzy connected if is a connected for for all .
(5) is called full intuitionistic fuzzy connected if is connected for all .
Proposition 63. If is connected, then is weakly connected.
Proof. connected implies that is connected. Now and so is weakly connected.
Proposition 64. If is firm and weakly connected, then is connected.
Proof. If is connected for some , then is connected since is firm.
Proposition 65. (1) If is a weak intuitionistic fuzzy tree, then is weakly connected and is a weak intuitionistic fuzzy forest. Conversely, if , with and such that is a forest and is connected, then is a weak intuitionistic fuzzy tree.
(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.
(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 , then is connected and is a forest. For the converse, we note that must also be a forest. Since also is connected, is a tree.
The proofs of (2), (3), and (4) are immediate.
Proposition 66. is firm if and only if is firm for all .
Proof. Suppose that is firm. Let . Let . Then
Hence and . Thus we conclude that and and is firm.
Conversely, suppose that is firm for all . Let , and let . Then . Now and since is firm and . Let . Then . Thus
Hence is firm.
5. 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of the paper.
Acknowledgment
The authors are thankful to the referees for their valuable comments and suggestions for improving the paper.