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In this paper, we determine operations on intuitionistic fuzzy graphs. The purpose is to apply the results to the combination of human trafficking routes. However, we use arbitrary dual [Formula: see text]-norms and [Formula: see text]-conorms to form these combinations since they determine more accurate models than maximum and minimum. We illustrate our results by combining two human trafficking routes from Central America to the United States.

This paper studies the closure and interior operators in LM-fuzzy topological spaces. The algebraic structures associated with various collections of closed sets and open sets are identified. Further, certain lattices formed by these algebraic structures are obtained and some lattice theoretic properties of the same are investigated. Corresponding to every element in M, the study associates a lattice of monoids which is determined by various types of closed sets and open sets.

This paper analyzes and modifies the concepts of degree of regularity and degree of normality introduced by Shi in 2011. Also the authors introduce extensions of [Formula: see text]-fuzzy topological spaces and explore the link between the associated [Formula: see text]-topologies of a given space and its extensions. Further, the effect of extensions on the degree of regularity and degree of normality of an [Formula: see text]-fts has been investigated.

This paper identifies a De Morgan algebra associated with the families of (r, s)-regular fuzzy open sets and (r, s)-regular fuzzy closed sets in a double fuzzy topological space. The situation under which this De Morgan algebra becomes a Boolean algebra is characterized. Certain other properties of this algebra are also investigated.

Connectivity parameters have important role in the study of networks in the physical world. Wiener index is one such parameter with several applications in chemistry and network theory. In this article Wiener index of various fuzzy graph structures like fuzzy trees and fuzzy cycles are discussed. Relationship between connectivity index and Wiener index of a fuzzy graph is also studied. For the effective computation of Wiener Index of a fuzzy graph an algorithm is proposed. A real life application related with human trafficking and illegal immigration is presented towards the end.

The algebraic structures of the families of fuzzy sets that arise out of various notions of openness and closedness in a double fuzzy topological space are investigated. The collection of these families forms a bounded, associative lattice. The zero divisors and zero-divisor graph of this lattice are also identified.

Super strongly perfect graphs and their association with certain other classes of graphs are discussed in this paper. It is observed that every split graph is super strongly perfect. An existing result on super strongly perfect graphs is disproved providing a counter example. It is also established that if a graph G contains a cycle of odd length, then its line graph L(G) is not always super strongly perfect. Complements of cycles of length six or above are proved to be non-super strongly perfect. If a graph is strongly perfect, then it is shown that they are super strongly perfect and hence all [Formula: see text]-free graphs are super strongly perfect.

Connectivity is a high priority concept in fuzzy graphs. The stability of a fuzzy graph is determined by the strength of connectedness of each pair of its nodes. This paper introduces two measures of connectivity called as connectivity index and average connectivity index of fuzzy graphs. Several results related to these parameters are obtained. These measures can be used to determine the potential flow of a fuzzy graph. Certain special types of nodes called CEN, CRN, and neutral nodes are also introduced and algorithms for finding these nodes are provided. Applications of these concepts in two different type of networks are provided.

Fractal theory is the study of irregularity which occurs in natural objects. It also enables us to see patterns in the highly complex and unpredictable structures resulting from many natural phenomena, using self-similarity property. The most common mathematical method to generate self-similar fractals is using an iterated function system (IFS). This paper discusses separation properties of finite products of hyperbolic IFSs. Characterizations for totally disconnected and overlapping product IFSs are obtained. A method to generate an open set which satisfies the open set condition for a totally disconnected IFS is given. Some necessary and sufficient conditions for a product IFS to be just touching are discussed. Also, Type 1 homogenous IFSs are introduced and its separation properties in terms of the separation properties of coordinate projections are explained towards the end.

In any network, the problem of reduction of flow is more important and frequent than the disconnection of the network. Problems related with maximum bandwidth paths and maximal flows are also very significant. In this article, the authors discuss connectivity concepts related with paths and flows in interconnection networks in a fuzzy framework. Some of the properties and applications of a stable structure called transitive block are discussed. Blocks in fuzzy graphs generalize the concept of blocks in graphs. The existence of a strongest strong cycle in any fuzzy block having at least three vertices is established. Four characterizations previously proposed for a class of blocks in fuzzy graphs are found to be true for a larger class. Two subclasses of θ-fuzzy graphs, namely, connectivity-transitive and cyclically-transitive fuzzy graphs are introduced to get a better understanding of blocks in fuzzy graphs. An application of transitive blocks in interconnection networks is also proposed.

We continue our study of connectedness properties of fuzzy graphs. Generalizations for some existing vertex and edge connectivity in graphs are presented. New parameters are compared with older ones and generalized values are calculated for some of the major classes like cycles and trees in fuzzy graphs. We consider illegal human trafficking flow between 10 different subregions in the world.

We turn our attention to the crimes of human trafficking and modern slavery. We rank countries with respect to their achievement of the SDGs pertinent to these crimes. We also use two fuzzy similarity measures to determine the similarity of the ranking methods.

We compare the rankings of the countries in their achievement of the SDGs by using the similarity measures discussed in Chap. 1 and below in Sect. 6.2. We take the average of the three rankings using the AHP, Guiasu, and Yen method to determine one ranking. We also take the average of the three averages determined by the Stakeholder method to obtain one ranking. We then determine the ranking obtained in [68] for our third ranking. We make our comparisons in Sect. 6.3 of these three rankings. In Sect. 6.4, we compare the rankings of the countries with respect to human trafficking.

This paper introduces extensions of a double fuzzy topological space and investigates the link between the associated fuzzy topologies of a given space and its extensions. It is also explored whether the \((r, s)\)-connectedness and \((p, q)\)-compactness of a double fuzzy topological space are carried over to its extensions.

Fuzzy relations are fundamental in applications of fuzzy set theory and fuzzy logic. The entire literature on fuzzy relations as applied to fuzzy graph theory are based on Rosenfeld’s relations. Rosenfeld used minimum and maximum as the norm and conorm in his study of compositions of fuzzy relations. In this paper, we generalize fuzzy relations using arbitrary [Formula: see text]-norms and [Formula: see text]-conorms. Many of the results do not hold when minimum and maximum are replaced by an arbitrary norm and an arbitrary conorm. Reflexive, symmetric and transitive generalized fuzzy relations are also discussed and an application to human trafficking and illegal immigration is presented.

Edges of a fuzzy graph are mainly classified into [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we study certain saturation counts with respect to the classification of edges. Characterizations for fuzzy cycles, fuzzy trees, blocks in fuzzy graphs and complete fuzzy graphs are also obtained using saturation counts.

The separation properties of self similar sets are discussed in this article. An open set condition for the (n, m)- iterated function system is introduced and the concepts of self similarity, similarity dimension and Hausdorff dimension of the attractor generated by an (n, m) - iterated function system are studied. It is proved that the similarity dimension and the Hausdorff dimension of the attractor of an (n, m) - iterated function system are equal under this open set condition. Further a necessary and sufficient condition for a set to satisfy the open set condition is established.

Directed fuzzy networks are introduced in this paper. They are normalized node capacitated networks and provide a good platform to model different types of complicated flows in nature. A directed fuzzy network version of Menger’s theorem and the celebrated Max flow Min cut theorem are also provided. Since the maximum flow through minimum number of directed internally disjoint paths is important in quality of service (QoS) problems in networking, the results in this paper can be applied to a wide variety of problems.

Trafficking in persons is the most heinous and condemnable organized crime of our time. The eradication of this crime must involve the full cooperation of all. In this paper, we provide a mathematical method to model such cooperation. We use a new concept in fuzzy graph theory, namely, that of incidence. We introduce fuzzy influence graph and characterize influence cutpairs in fuzzy influence graphs since their removal increases the number of connected components of a fuzzy network and thus weakens the potential flow in the network. Some of the properties of influence pairs and influence cutnodes are also studied.

Intervals and convexity play crucial roles in the applications of graph theory such as town planning and design of graphics. In this article, the concept of geodetic interval in graphs is extended to fuzzy graphs. Intervals are useful in the study of properties of fuzzy graphs which depend on the geodetic distance between vertices. The axiomatic definition of intervals in fuzzy graphs are used to define intervals in different fuzzy graph structures like fuzzy trees and complete fuzzy graphs. Finally a set theoretic operations of intervals like union, intersection are also discussed and some results are obtained.

Connectivity concepts in fuzzy incidence graphs are discussed in this article. Fuzzy incidence graphs are important in interconnection networks with influenced flows. Thus it is of primary importance to analyze their connectivity properties. The existence of a strong path between any node arc pair of a fuzzy incidence graph is established. Cut pairs and fuzzy incidence trees are characterized using the concept of strong pairs. Several other structural properties of fuzzy incidence graphs are also obtained.

The notion of a fuzzy end node in fuzzy incidence graphs is developed in this article. The concept of a fuzzy incidence graph is new and should be very useful in applications in fuzzy graphs and fuzzy networks. Relationships between weak incidence pairs and fuzzy incidence end nodes and fuzzy incidence cut vertices are developed.

In this paper, we prove some fixed point theorems for ψF-contractions in the framework of quasi-metric spaces generalizing and improving several similar results in metric spaces. At the same time, we consider iterated function systems consisting of ψF-contractions on quasi-metric spaces, and we give some sufficient conditions for the existence and uniqueness of their attractor which is, generally, a fractal. Some illustrative examples are provided.

Connectivity in fuzzy incidence graphs (FIG) is studied in this article. Different connectivity aspects of fuzzy incidence graphs such as bonding pair, doubly bonding pair and incidence cut of pairs are discussed. Incidence connectivity and incidence connectivity of pairs are introduced and results similar to Whitney’s Theorem are presented. The concept of t-connected fuzzy incidence graphs are also studied and some characterizations are obtained. An application related with illegal migration is presented. The most vulnerable routes in the Mexican-US border are focussed on and corresponding risks are evaluated using t-conorms.

Two major parameters associated with fuzzy graphs are discussed in this chapter. The first one, named as connectivity index gives an indication about the flow through the network and the second, called as Wiener index gives the average distance.

In this section, we introduce the concept of fuzzy geodetic convex sets and discuss some of their properties.

Fractal geometry is one of the beautiful and challenging branches of mathematics. Self similarity is an important property, exhibited by most of the fractals. Several forms of self similarity have been discussed in the literature. Iterated Function System (IFS) is a mathematical scheme to generate fractals. There are several variants of IFSs such as condensation IFS, countable IFS, etc. In this paper, certain properties of self similar sets, using the concept of boundary are discussed. The notion of boundaries like similarity boundary and dynamical boundary are extended to condensation IFSs. The relationships and measure theoretic properties of boundaries in dynamical systems are analyzed. Self similar sets are characterized using the Hausdorff measure of their boundaries towards the end.

Fractal geometry is mainly based on the idea of self-similar forms. To be self-similar, a shape must able to be divided into parts that are smaller copies, which are more or less similar to the whole. There are different forms of self similarity in nature and mathematics. In this paper, some of the topological properties of super self similar sets are discussed. It is proved that in a complete metric space with two or more elements, the set of all non super self similar sets are dense in the set of all non-empty compact sub sets. It is also proved that the product of self similar sets are super self similar in product metric spaces and that the super self similarity is preserved under isometry. A characterization of super self similar sets using contracting sub self similarity is also presented. Some relevant counterexamples are provided. The concepts of exact super and sub self similarity are introduced and a necessary and sufficient condition for a set to be exact super self similar in terms of condensation iterated function systems (Condensation IFS’s) is obtained. A method to generate exact sub self similar sets using condensation IFS’s and the denseness of exact super self similar sets are also discussed.

The main purpose of this paper is to introduce weak isomorphism, co-weak isomorphism and isomorphism of neighborly irregular vague graphs. Some results on order, size and degree of nodes in isomorphic neighborly irregular and isomorphic highly irregular vague graphs are discussed. Isomorphism between neighborly irregular and highly irregular vague graphs are proved to be an equivalence relation. Density and balanced irregular vague graphs are introduced. Finally, an application of vague graphs is given.

Certain properties of the lattices of L-topologies determined by the families of Scott continuous functions for a given topological space are investigated. The authors disprove certain known theorems on the above lattices and the correct results are provided using the concept of strict chain join-generability in lattices.

Recall that if x is a member of a set X and \(t\in [0,1],\) then we define the fuzzy subset \(x_{t}\) of X by \(x_{t}(x)=t\) and \(x_{t}(y)=0\) for all \(y\in X\backslash \{x\}.\) We call \(x_{t}\) a fuzzy singleton

The results of the next two sections are based on Somasundaram, Somasundaram (Pattern Recongit Lett 19:787–791, 1998), [29]. The formal mathematical definition of domination was given by Ore, (Theory of graphs. American Mathematical Society, Providence, 1962), [22]. Cockayne and Hedetnieme, (Networks 7:247–261, 1977, [3]), published a survey paper on this topic in 1977 and since then hundreds of papers have been published on this subject. According to Somasundaram, Somasundaram (Pattern Recongit Lett 19:787–791, 1998, [29]), the rapid growth of research in this area is due to the following three factors. (1) The diversity of applications of domination theory to both real world and mathematical coverings or location problems. (2) The wide variety of domination parameters that can be defined. (3) The NP-completeness of the basic domination problem, its close and natural relationship to other NP-complete problems and the subsequent interest in finding polynomial time solutions to domination problems in special classes of graphs.

We introduce the notion of the degree of incidence of a vertex and an edge in a fuzzy graph in fuzzy graph theory. We concentrate on incidence, where the edge is adjacent to the vertex. We determine results concerning bridges, cutvertices, cutpairs, fuzzy incidence paths, fuzzy incidence tree for fuzzy incidence graphs. In (Dinesh, Ph.D. thesis, Kannur University, Kerala, India, [4]; Dinesh, Adv Fuzzy Sets Syst, 21:33–48, 2016, [5]), Dinesh introduced the notion of the degree of incidence of a vertex and an edge in fuzzy graph theory. This notion seem to have potential use in a variety of areas involving networks, Mordeson (Adv Fuzzy Sets Syst, 22:121–133, 2016, [12]).

In this chapter, we discuss three different concepts in fuzzy graphs. The first concept is that of sequences of fuzzy graphs, which allow us to connect a fuzzy graph to a sequence space. Most of the fuzzy graph structures are characterized using different types of sequences. In the second part of this chapter, we discuss saturation in fuzzy graphs and the third part deals with strong intervals and strong gates in fuzzy graphs.