Applications of Fuzzy Graphs

Types of arcs in a fuzzy graph

Privacy preserving network publishing has been studied extensively in recent years. Although more works have adopted un-weighted graphs to model network relationships, weighted graph modeling can provide deeper analysis of the degree of relationships. Previous works on weighted graph privacy have concentrated on preserving the shortest path characteristic between pairs of vertices. Two common types of privacy have been proposed. One type of privacy tried to add random noise edge weights to the graph but still maintain the same shortest path. The other privacy, k-shortest path privacy, minimally perturbed edge weights so that there exists k shortest paths. However, the k-shortest path privacy only considers anonymizing same fixed number of shortest paths for all pairs of source and destination vertices. In this work, we present a new concept called [k1, k2]-shortest path privacy to allow different number of shortest paths for different pairs of vertices. A published network graph with [k1, k2]-shortest path privacy has at least k' indistinguishable shortest paths between the source and destination vertices, where k1 l k' l k2. A heuristic algorithm based on modifying only Non-Visited (NV) edges is proposed and experimental results showing the feasibility and characteristics of the proposed approach are presented.

We introduce a model for differentially private analysis of weighted graphs in which the graph topology (υ,e) is assumed to be public and the private information consists only of the edge weights ω : e → R+. This can express hiding congestion patterns in a known system of roads. Differential privacy requires that the output of an algorithm provides little advantage, measured by privacy parameters e and δ, for distinguishing between neighboring inputs, which are thought of as inputs that differ on the contribution of one individual. In our model, two weight functions w,w' are considered to be neighboring if they have l1 distance at most one.We study the problems of privately releasing a short path between a pair of vertices and of privately releasing approximate distances between all pairs of vertices. We are concerned with the approximation error, the difference between the length of the released path or released distance and the length of the shortest path or actual distance.For the problem of privately releasing a short path between a pair of vertices, we prove a lower bound of Ω(|υ|) on the additive approximation error for fixed privacy parameters e,δ. We provide a differentially private algorithm that matches this error bound up to a logarithmic factor and releases paths between all pairs of vertices, not just a single pair. The approximation error achieved by our algorithm can be bounded by the number of edges on the shortest path, so we achieve better accuracy than the worst-case bound for pairs of vertices that are connected by a low-weight path consisting of o(|υ|) vertices.For the problem of privately releasing all-pairs distances, we show that for trees we can release all-pairs distances with approximation error $O(log2.5|υ|) for fixed privacy parameters. For arbitrary bounded-weight graphs with edge weights in [0,M] we can brelease all distances with approximation error O(√>(|υ|M).

Nowadays the fuzzy graphs gained popularity due to their wide applications in different areas of science, engineering, social sciences, etc.In this paper, we consider covering problem in fuzzy graph. Different types of covering problems have been defined in this paper. Almost all problems are new. For each of these problems separate study is required. Also, the covering set, strong covering set, minimal covering set, etc., are defined and explained with examples. In a graph, each vertex can cover only those vertices that lie within its covering radius along shortest path and no vertex can cover itself. Here, an algorithm is designed to find the different types of covering sets on a fuzzy graph. An imprecise number, instead of a real number, namely interval number and triangular fuzzy number are considered as arc length of a fuzzy graph. To the best of our knowledge no works are available for covering problem on fuzzy graphs/networks. An application of covering set in natural disaster management is discussed to highlight the importance of the covering problem. In this application, an algorithm is designed to find all the facility vertices (or supply vertices) for given vertex and covering radius.

Point-to-point shortest path (PPSP), or s-t connectivity, is a variant of the shortest path problem found in graph theory. In this problem, we are given a graph and pairs of vertices over time, and the output is the shortest path between each pair of vertices. In this paper, we present two algorithms. Our first algorithm approximately solves the PPSP problem on any arbitrary graph. For each pair of vertices queried, it accurately and efficiently estimates the shortest path between the two vertices. Our second algorithm extends the first to work on dynamic graphs. That is, our second algorithm can efficiently account for changes in the graph, such as friend requests on the Facebook network or road closures on road networks. At a high level, both algorithms partition the graph into highly connected communities. To respond to a query q(u, v), they each find the fewest number of partitions between u and v, and the shortest path through each partition. We show that our static graph algorithm can approximate the distance between two vertices with about 20%-35% percent error and anywhere from 80X-70000X faster than a BFS in practice with the right choice of partitions. Additionally, we show that our dynamic graph algorithm can account for updates to the graph anywhere from 20X-20000X faster than rerunning the static graph algorithm for each change to the graph.

A graph is a pair (V, R), where V is a set and R is a relation on V. The elements of V are thought of as vertices of the graph and the elements of R are thought of as the edges Similarly, any fuzzy relation ρ on a fuzzy subset μ of a set V can be regarded as defining a weighted graph, or fuzzy graph, where the edge (x, y) ∈ V × V has weight or strength ρ(x, y) ∈ [0, 1]. In this chapter, we shall use graph terminology and introduce fuzzy analogs of several basic graph-theoretical concepts. For simplicity, we will consider only undirected graphs through out this chapter unless otherwise specified. Therefore, all fuzzy relations are symmetric and all edges are regarded as unordered pairs of vertices. We abuse notation by writing (x, y) for an edge in an undirected graph (V, R), where x, y ∈ V. (We need not consider loops, that is, edges of the form (x, x); we can assume, if we wish, that our fuzzy relation is reflexive.) Formally, a fuzzy graph G = (V, μ, ρ) is a nonempty set V together with a pair of functions μ: V → [0, 1] and μ: V × V → [0, 1] such that for all x, y in V, ρ(x, y) ≤ μ(x) ∧ μ(y). We call μ the fuzzy vertex set of G and ρ the fuzzy edge set of G, respectively. Note that ρ is a fuzzy relation on μ. We will assume that, unless otherwise specified, the underlying set is V and that it is finite. Therefore, for the sake of notational convenience, we omit V in the sequel and use the notation G = (μ,ρ). Thus in the most general case, both vertices and edges have membership value. However, in the special case where μ(x) = 1, for all x ∈ V, edges alone have fuzzy membership. So, in this case, we use the abbreviated notation G = (V, ρ). The fuzzy graph H = (v,τ) is called a partial fuzzy subgraph of G = (μ, ρ) if v ⊆ μ and τ ⊆ ρ. Similarly, the fuzzy graph H = (P, v, τ) is called a fuzzy subgraph of G = (V, μ, ρ) induced by P if P ⊆ V, v(x) = μ(x) for all x ∈ P and τ(x, y) = ρ(x, y) for all x, y ∈ P. For the sake of simplicity, we sometimes call H a fuzzy subgraph of G. It is worth noticing that a fuzzy subgraph (P, v, τ) of a fuzzy graph (V, μ, ρ) is in fact a special case of a partial fuzzy subgraph obtained as follows.KeywordsIncidence MatrixIntersection GraphFuzzy SubsetInterval GraphFuzzy RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

An application of fuzzy graph in accidental prone zone to reduce the traffic congestion

Today, fuzzy graphs (FGs) have a variety of applications in other fields of study, including medicine, engineering, and psychology, and for this reason, many researchers around the world are trying to identify their properties and use them in computer sciences as well as finding the smallest problem in a network. The concept of a Cayley fuzzy graph has become a standard part of the toolkit used to investigate and describe groups. Also, Cayley fuzzy graphs are good models for interconnection networks, and they are useful in semigroup theory for establishing which elements are ℓ and R related. The previous definition limitations in the FGs have directed us to offer a new classification in terms of Cayley fuzzy graphs. So, in this paper, two new definitions of Cayley fuzzy graphs (CFGs) and pseudo-Cayley fuzzy graphs (PCFGs) are discussed and their rough approximations are studied. Also, some properties of fuzzy rough sets (FRSs) in CFGs and PCFGs have been investigated. Finally, we presented the determination of the most effective person in the Water and Sewerage Organization and the importance of using refereeing facilities in football matches between club teams, by using CFG in the presented applications.

We describe cellular-automaton-based algorithms for solving two shortest path problems on arbitrary connected, directed, and weighted graphs with n vertices. The first problem is that of finding the shortest path from a given vertex to another given vertex of the graph. A two-dimensional cellular automaton, shaped as a triangle, with \(O(n^2)\) cells, is used. The algorithm runs in O(n) time. The second problem requires that all shortest paths between pairs of vertices be obtained. An \(n \times n\) cellular automaton solves the problem in \(O(n \log n)\) time.

The growing popularity of online Earth visualization tools and geo-realistic games and the availability of high resolution terrain data have motivated a new class of queries to the interests of the GIS and spatial database community: spatial queries (e.g., kNN) over land surface. However, the fundamental challenges that restrict the applicability of these studies to real world applications are the prohibitive time complexity and storage overhead to precompute the shortest surface paths. In this paper, for the first time, we propose an approximate solution to address both challenges and allow browsing the shortest surface paths in O(log N + √N) time, where N is the size of the terrain. With this method, the time and space requirements for an exhaustive all-pair pre-computation have been reduced from O(N3) to O(N1.5) and O(N) respectively. The substantial savings in both time and storage are gained by taking advantage of the fact that the O(N2) surface paths only deviate from approximate straight lines at O(√N) points, termed rough vertices. As a result, we propose a linear time shortest surface path computation algorithm between two arbitrary vertices and a linear size storage structure, which captures all the shortest surface paths between any pair of vertices. We experimentally verified the applicability and scalability of the proposed methods with large real world and synthetic data sets and showed that accuracy higher than 97% can be obtained in most cases.

This paper discusses fuzzy labeling graph and its properties. The properties discussed are fuzzy labeling subgraph, union graph, and fuzzy magic graph. The results showed that the strength of connectedness for pair of vertices in a graph with fuzzy labeling is always greater than or equal to the strength of connectedness for pair of vertices in its subgraph. In addition, union of two graph with fuzzy labeling will form a graph with fuzzy labeling. Meanwhile, on magic labeling there is always a fuzzy bridge on fuzzy magic graph.

This paper describes the theory and algorithms of fuzzy distance transform (FDT). Fuzzy distance is defined as the length of the shortest path between two points. The length of a path in a fuzzy subset is defined as the integration of fuzzy membership values of the points along the path. The shortest path between two points is the one with the minimum length among all (infinitely many) paths between the two points. It is demonstrated that, unlike in the binary case, the shortest path in a fuzzy subset is not necessarily a straight-line segment. The support of a fuzzy subset is the set of points with nonzero membership values. It is shown that, for any fuzzy subset, fuzzy distance is a metric for the interior of its support. FDT is defined as the process on a fuzzy subset that assigns at each point the smallest fuzzy distance from the boundary of the support. The theoretical framework of FDT in continuous space is extended to digital spaces and a dynamic programming-based algorithm is presented for its computation. Several potential medical imaging applications are presented including the quantification of blood vessels and trabecular bone thickness in the regime of limited special resolution.

For a simple graph [Formula: see text] and for a pair of vertices [Formula: see text], we say that a vertex [Formula: see text] resolves [Formula: see text] and [Formula: see text] if the shortest path from [Formula: see text] to [Formula: see text] is of a different length than the shortest path from [Formula: see text] to [Formula: see text]. A set of vertices [Formula: see text] is a resolving set if for every pair of vertices [Formula: see text] and [Formula: see text] in [Formula: see text], there exists a vertex [Formula: see text] that resolves [Formula: see text] and [Formula: see text]. The minimum weight resolving set problem is to find a resolving set [Formula: see text] for a weighted graph [Formula: see text] such that [Formula: see text] is minimum, where [Formula: see text] is the weight of vertex [Formula: see text]. In this paper, we explore the possible solutions of this problem for grid graphs [Formula: see text] where [Formula: see text]. We give a complete characterization of solutions whose cardinalities are 2 or 3, and show that the maximum cardinality of a solution is [Formula: see text]. We show that the grid has the property that given a landmark set, we only need to investigate whether or not all pairs of vertices that share common neighbors are resolved to determine if the whole graph is resolved. We use this result to provide a characterization of a class of minimals whose cardinalities range from [Formula: see text] to [Formula: see text] and show that the number of such minimals is [Formula: see text].

The product fuzzy graph is an extension of the fuzzy graph definition by replacing the minimum operation with the product operation. This research is qualitative research with research stages consisting of determining open problems, constructing new definitions, constructing new properties as theorems, and verifying results in the form of mathematical proofs. The purpose of this research is to get the Cartesian product definition of the product fuzzy graph and its properties. The results showed that the Cartesian product of two product fuzzy graphs is a fuzzy graph, the Cartesian product of two strong product fuzzy graphs is a strong product fuzzy graph, and the resulting Cartesian product of two complete product fuzzy graphs is a strong product fuzzy graph. Thus, the novelty of this research is that new properties of product fuzzy graphs are obtained.

Longest paths and cycles in faulty star graphs