In this article, the parabolic dense fuzzy set is defined, and its basic arithmetic operations ar... more In this article, the parabolic dense fuzzy set is defined, and its basic arithmetic operations are studied with graphical illustration. The lock set concept is incorporated in a parabolic dense fuzzy set. Then, it is applied to the problems of fishery culture via the modeling of an economic order quantity model. Here, the fingerlings are fed to reach the ideal size to fulfill the customer’s demand. The growth rate of the fingerlings is assumed as a linear function. After the sales of all fish, the pond is cleaned properly for a new cycle. Here, the model is solved in a crisp sense first. Then, we fuzzify the model considering the demand rate as a parabolic dense lock fuzzy number and obtain the result in a fuzzy environment. The main aim of our study was to find the quantity of the ordering items such that the total inventory cost gets a minimum value. Lastly, sensitivity analysis and graphical illustrations were added for better justification of our model.
In this study, the notions of picture fuzzy tolerance graphs, picture fuzzy interval containment ... more In this study, the notions of picture fuzzy tolerance graphs, picture fuzzy interval containment graphs and picture fuzzy $$\phi $$ ϕ -tolerance graphs are established. Three special types of picture fuzzy tolerance graphs having bounded representations are introduced and studied corresponding properties of them taking $$\phi $$ ϕ as max, min and sum functions. Also, picture fuzzy proper and unit tolerance graphs are established and some related results are investigated. The class of picture fuzzy $$\phi $$ ϕ -tolerance chaingraphs which is the picture fuzzy $$\phi $$ ϕ -tolerance graphs of a nested family of picture fuzzy intervals are presented. A real-life application in sports competition is modeled using picture fuzzy min-tolerance graph. Also a comparison is given between picture fuzzy tolerance graphs and intuitionistic fuzzy tolerance graphs.
In an undirected graph, the feedback vertex set problem is to find a set of vertices of minimum c... more In an undirected graph, the feedback vertex set problem is to find a set of vertices of minimum cardinality whose removal makes the graph acyclic (connected or disconnected). This problem is known to be NP-hard for general graph. In this paper, we proposed two algorithms for finding the minimum feedback vertex set on interval graphs. The first algorithm is for unweighted case and takes O(n + m) time while the second one is for weighted case taking O(n p logC) time where n is the number of vertices, m is the number of edges and C is the cost of the longest path of the graph.
ABSTRACT In this paper, we define fuzzy tolerance, fuzzy tolerance graph, fuzzy bounded tolerance... more ABSTRACT In this paper, we define fuzzy tolerance, fuzzy tolerance graph, fuzzy bounded tolerance graph, fuzzy interval containment graph and regular representation of fuzzy tolerance graph, fuzzy unit tolerance graph and proper tolerance graph. Also some basic theorems related to the stated graphs have been presented.
Abstract Vague graph is a generalized structure of fuzzy graph which gives more precision, flexib... more Abstract Vague graph is a generalized structure of fuzzy graph which gives more precision, flexibility, and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, we introduced the notion of regular, totally regular product vague graphs, and product vague line graph. We proved that under some conditions regular and totally regular product vague graph becomes equivalent. Some properties of product vague line graph are investigated. We showed that a product vague graph is isomorphic to its corresponding product vague line graph under some conditions.
Http Dx Doi Org 10 1080 00207160412331286851, Jan 25, 2007
ABSTRACT A spanning tree T is said to be a tree t-spanner of a graph G if the distance between an... more ABSTRACT A spanning tree T is said to be a tree t-spanner of a graph G if the distance between any two vertices in T is at most t times their distance in G. The tree t-spanner has many applications in network and distributed environments. This problem is NP-hard for general graph even for some special classes of graphs. This problem is polynomial solvable for interval graph when t ≥ 3. When t = 2 the exact complexity of the problem still remains open, but, for t = 2 a polynomial time 2-approximation algorithm is available. An O(n + m) time sequential algorithm is available to solve tree 3-spanner problem, where m and n, respectively, represent the number of edges and the number of vertices of the interval graph. Here, a parallel algorithm is designed to solve a tree 3-spanner problem in O(log n) time using O (n/log n) processors on an EREW PRAM. As a byproduct, a parallel algorithm is also designed to find the increasing sequence of numbers of a set of N numbers in O (log N) time using O(N/log N) processors.
Intersection graphs are very important in both theoretical as well as application point of view. ... more Intersection graphs are very important in both theoretical as well as application point of view. Depending on the geometrical representation, different type of intersection graphs are defined. Among them interval, circular-arc, permutation, trapezoid, chordal, disk, circle graphs are more important. In this article, a brief introduction of each of these intersection graphs is given. Some basic properties and algorithmic status of few problems on these graphs are cited. This article will help to the beginners to start work in this direction. Since the article contains a lot of information in a compact form it is also useful for the expert researchers too.
In this article, the parabolic dense fuzzy set is defined, and its basic arithmetic operations ar... more In this article, the parabolic dense fuzzy set is defined, and its basic arithmetic operations are studied with graphical illustration. The lock set concept is incorporated in a parabolic dense fuzzy set. Then, it is applied to the problems of fishery culture via the modeling of an economic order quantity model. Here, the fingerlings are fed to reach the ideal size to fulfill the customer’s demand. The growth rate of the fingerlings is assumed as a linear function. After the sales of all fish, the pond is cleaned properly for a new cycle. Here, the model is solved in a crisp sense first. Then, we fuzzify the model considering the demand rate as a parabolic dense lock fuzzy number and obtain the result in a fuzzy environment. The main aim of our study was to find the quantity of the ordering items such that the total inventory cost gets a minimum value. Lastly, sensitivity analysis and graphical illustrations were added for better justification of our model.
In this study, the notions of picture fuzzy tolerance graphs, picture fuzzy interval containment ... more In this study, the notions of picture fuzzy tolerance graphs, picture fuzzy interval containment graphs and picture fuzzy $$\phi $$ ϕ -tolerance graphs are established. Three special types of picture fuzzy tolerance graphs having bounded representations are introduced and studied corresponding properties of them taking $$\phi $$ ϕ as max, min and sum functions. Also, picture fuzzy proper and unit tolerance graphs are established and some related results are investigated. The class of picture fuzzy $$\phi $$ ϕ -tolerance chaingraphs which is the picture fuzzy $$\phi $$ ϕ -tolerance graphs of a nested family of picture fuzzy intervals are presented. A real-life application in sports competition is modeled using picture fuzzy min-tolerance graph. Also a comparison is given between picture fuzzy tolerance graphs and intuitionistic fuzzy tolerance graphs.
In an undirected graph, the feedback vertex set problem is to find a set of vertices of minimum c... more In an undirected graph, the feedback vertex set problem is to find a set of vertices of minimum cardinality whose removal makes the graph acyclic (connected or disconnected). This problem is known to be NP-hard for general graph. In this paper, we proposed two algorithms for finding the minimum feedback vertex set on interval graphs. The first algorithm is for unweighted case and takes O(n + m) time while the second one is for weighted case taking O(n p logC) time where n is the number of vertices, m is the number of edges and C is the cost of the longest path of the graph.
ABSTRACT In this paper, we define fuzzy tolerance, fuzzy tolerance graph, fuzzy bounded tolerance... more ABSTRACT In this paper, we define fuzzy tolerance, fuzzy tolerance graph, fuzzy bounded tolerance graph, fuzzy interval containment graph and regular representation of fuzzy tolerance graph, fuzzy unit tolerance graph and proper tolerance graph. Also some basic theorems related to the stated graphs have been presented.
Abstract Vague graph is a generalized structure of fuzzy graph which gives more precision, flexib... more Abstract Vague graph is a generalized structure of fuzzy graph which gives more precision, flexibility, and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, we introduced the notion of regular, totally regular product vague graphs, and product vague line graph. We proved that under some conditions regular and totally regular product vague graph becomes equivalent. Some properties of product vague line graph are investigated. We showed that a product vague graph is isomorphic to its corresponding product vague line graph under some conditions.
Http Dx Doi Org 10 1080 00207160412331286851, Jan 25, 2007
ABSTRACT A spanning tree T is said to be a tree t-spanner of a graph G if the distance between an... more ABSTRACT A spanning tree T is said to be a tree t-spanner of a graph G if the distance between any two vertices in T is at most t times their distance in G. The tree t-spanner has many applications in network and distributed environments. This problem is NP-hard for general graph even for some special classes of graphs. This problem is polynomial solvable for interval graph when t ≥ 3. When t = 2 the exact complexity of the problem still remains open, but, for t = 2 a polynomial time 2-approximation algorithm is available. An O(n + m) time sequential algorithm is available to solve tree 3-spanner problem, where m and n, respectively, represent the number of edges and the number of vertices of the interval graph. Here, a parallel algorithm is designed to solve a tree 3-spanner problem in O(log n) time using O (n/log n) processors on an EREW PRAM. As a byproduct, a parallel algorithm is also designed to find the increasing sequence of numbers of a set of N numbers in O (log N) time using O(N/log N) processors.
Intersection graphs are very important in both theoretical as well as application point of view. ... more Intersection graphs are very important in both theoretical as well as application point of view. Depending on the geometrical representation, different type of intersection graphs are defined. Among them interval, circular-arc, permutation, trapezoid, chordal, disk, circle graphs are more important. In this article, a brief introduction of each of these intersection graphs is given. Some basic properties and algorithmic status of few problems on these graphs are cited. This article will help to the beginners to start work in this direction. Since the article contains a lot of information in a compact form it is also useful for the expert researchers too.
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