Tolerance Graphs Research Articles

Tolerance fuzzy graphs

Tolerance graphs were introduced in 1982 by M. C. Golumbic and C. L. Monma as a generalization of interval graphs. In this paper, we introduce tolerance fuzzy graphs as a generalization of tolerance graphs, and apply them to a modeling of a transmission of airborne diseases.

UTILIZING THE EMBODIMENT FUNCTION RELATION AND TOLERANCE MODEL FOR ROBUST CONCEPT DESIGN

AbstractThe early use of Robust Design (RD) supports the development of product concepts with low sensitivity to variation, which offers advantages for reducing the risk of costly iterations. Due to the lack of approaches for early evaluation of product robustness, the embodiment-function-relation and tolerance (EFRT-) model was developed, which combines the contact and channel approach and tolerance graphs. The information exchange of both approaches offers a high potential for reliable robustness evaluation results. However, that potential currently relies unused, since the link between applicable robustness criteria and the extended information is missing. To solve this problem, four research steps were determined: (1) understanding of robustness, (2) collection of RD principles, (3) identification of EFRT-model information and (4) mapping of RD principles and information. The results show nine adapted RD principles, the identified model information for the robustness evaluation, the evaluation criteria as well as their mapping. Utilizing the mapping and the proposed criteria in this contribution, a more comprehensive robustness evaluation in early stages is enabled.

An investigation on m-polar fuzzy tolerance graph and its application

An investigation on m-polar fuzzy tolerance graph and its application

Picture fuzzy tolerance graphs with application

In this study, the notions of picture fuzzy tolerance graphs, picture fuzzy interval containment graphs and picture fuzzy phi -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 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 -tolerance chaingraphs which is the picture fuzzy phi -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.

Query-competitive sorting with uncertainty

We study the problem of sorting under incomplete information, when queries are used to resolve uncertainties. Each of n data items has an unknown value, which is known to lie in a given interval. We can pay a query cost to learn the actual value, and we may allow an error threshold in the sorting. The goal is to find a nearly-sorted permutation by performing a minimum-cost set of queries.We show that an offline optimum query set can be found in polynomial time, and that both oblivious and adaptive problems have simple query-competitive algorithms. The query-competitiveness for the oblivious problem is n for uniform query costs, and unbounded for arbitrary costs; for the adaptive problem, the ratio is 2.We then present a unified adaptive strategy for uniform query costs that yields the following improved results: (i) a 3/2-query-competitive randomized algorithm; (ii) a 5/3-query-competitive deterministic algorithm if the dependency graph has no 2-components after some preprocessing, which has query-competitive ratio 3/2+O(1/k) if the components obtained have size at least k; and (iii) an exact algorithm if the intervals constitute a laminar family. The first two results have matching lower bounds, and we have a lower bound of 7/5 for large components.We also give a randomized adaptive algorithm with query-competitive factor 1+433≈1.7698 for arbitrary query costs, and we show that the 2-query competitive deterministic adaptive algorithm can be generalized for queries returning intervals and for a more general graph problem (which is also a generalization of the vertex cover problem), by using the local ratio technique. Furthermore, we prove that the advice complexity of the adaptive problem is ⌊n/2⌋ if no error threshold is allowed, and ⌈n/3⋅lg⁡3⌉ for the general case.Finally, we present some graph-theoretical results regarding co-threshold tolerance graphs, and we discuss uncertainty variants of some classical interval problems.

Application of Tolerance Graphs to Combat COVID-19 Pandemic.

Tolerance graphs were introduced in 1982 by Golumbic and Monma as a generalization of interval graphs. In this paper, we propose several applications of tolerance graphs in fighting COVID-19. These applications include finding cliques of a certain size, and calculating the chromatic number of a graph, the problems that are in general NP-complete but for tolerance graphs can be solved in polynomial time.

Topological Structure of Generalized Rough Graphs

The main purpose of this paper, is to introduce a topological space , which is induced by reflexive graph and tolerance graph , such that may be infinite. Furthermore, we offered some properties of such as connectedness, compactness, Lindelöf and separate properties. We also study the concept of approximation spaces and get the sufficient and necessary condition that topological space is approximation spaces.

Interoperability of CAD geometry and product manufacturing information for computer integrated manufacturing

ABSTRACT Model-Based 3D engineering has gained wide popularity in recent times for various Computer-Aided Engineering applications, bypassing the need for 2D drawings, thereby creating an efficient digital thread, reducing development time and improving the product quality. With the use of model-based definition, most data required to design, analyze and manufacture aproduct is stored in digital format. The primary focus of the paper is to create aseamless digital thread for exchanging data from anative CAD file to astandard data format like ISO STEP AP242 and vice versa, which enables smart manufacturing by attaching product manufacturing information. Prior to STEP AP242, the only successful translation among CAD systems was for the boundary representation with very little or no manufacturing data. These standards will not be enough unless defined in amachine-readable format and are equipped with tools that can read/write from these data structures. The method presented enhances the neutral data format ‘Constraint Tolerance Feature (CTF) Graph’ to ‘Augmented Constraint Tolerance (aCTF) Graph’, which holds additional information required for manufacturing. This paper discusses the elements needed during CAD data exchange, for multiple CAE applications like Computer-Aided Tolerance Analysis, Computer-Aided Manufacturing, etc., and discusses the implementation of required translators.

Min-Orderable Digraphs

We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, complements of threshold tolerance graphs (known as co-TT graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs. (The last three classes coincide, but have been investigated in different contexts.) We show that all of the above classes are united by a common ordering characterization, the existence of a min ordering. However, because the presence or absence of reflexive relationships (loops) affects whether a graph or digraph has a min ordering, to obtain this result, we must define the graphs and digraphs to have those loops that are implied by their definitions. These have been largely ignored in previous work. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, characterized by the existence of a compact representation, a signed-interval model, which is a generalization of known representations of the graph classes. We show that the signed-interval digraphs are precisely those digraphs that are characterized by the existence of a min ordering when the loops implied by the model are considered part of the graph. We also offer an alternative geometric characterization of these digraphs. We show that co-TT graphs are the symmetric signed-interval digraphs, the adjusted interval digraphs are the reflexive signed-interval digraphs, and the interval graphs are the intersection of these two classes, namely, the reflexive and symmetric signed-interval digraphs. Similar results hold for bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs.

Pre-drill pore pressure prediction and safe well design on the top of Tulamura anticline, Tripura, India: a comparative study

The Tulamura anticline falls in the state Tripura, Northeast India. The anticline is extended up to neighbour country Bangladesh. The region is characterized by huge anticlines, normal faults and abnormally pressured formations which causes a wide margin of uncertainties in wildcat well planning and design. These geological complexities of Tulamura anticline make the drilling engineers more challenging. Therefore, a proper well design is essential in such a region to prevent blowout. Drilling engineer requires to maintain wellbore pressure between the pore pressure and fracture pressure to reduce the possibility of a kick and a formation damage. Pore pressure plays an important role to design a safe and economical well in such a high pressure and temperature reservoir. For wildcat drilling, only seismic data are available in the study area. There are various methods to predict pore pressure from seismic velocity data. Modified Eaton’s method is widely used for the pore pressure prediction from seismic data in terms of the velocity ratio. Modified Eaton’s equations may cause an error by manual selection of compaction trend line which is used to find normal compaction velocity. The main objectives of this study are to develop a new method to predict pore pressure and safe well design on the top of Tulamura anticline in terms of pore pressure. The new method is validated by a well-known method, modified Eaton’s method, and RFT pressure data from offset wells. An excellent match with pore pressures estimated from RFT pressure data and predicted by new model along with modified Eaton’s method is observed in this research work. The efficiency and accuracy level of the hybrid model is more as compared to other methods as it does not require compaction velocity data; thus, an error caused by manual compaction trend can be eliminated. Pore pressure predicted by new method indicates result up to the 6000 m, which is up to the basement rock. The predicted pore pressures by new method are used as an input to calculate the fracture pressure by Hubbert and Willis method, Mathews and Killy method and modified Eaton’s method. Equivalent mud weight selection is carried out using median line principle with additional 0.3 ppg, 0.3 ppg and 0.2 ppg of swab pressure, surge pressure and safety factor, respectively, for calculation of all casing pipes. Casing setting depths are selected based on pore pressure gradient, fracture pressure gradient and mud weight using graphical method. Here, four types of casing setting depths are selected: conductor, surface, intermediate and production casings at 100 ft, 6050 ft, 15500 ft and 18,500 ft, respectively, by new methods, but the casing setting depths for intermediate are at 13500 ft in the case of modified Eaton’s method. The casing policy is selected based on burst pressure, collapse pressure and tension load. For each casing, kick tolerance in bbl is determined from kick tolerance graph to prevent the blowout. Finally, comparative safe and economical wells are designed on the top of Tulamura anticline along with target depth selection, casing setting depth selection, casing policy selection and kick tolerance in consideration of collapse pressure, burst pressure and tension load which gives a clear picture of well planning on the top of anticline in pore pressure point of view.

A Branch & Price algorithm for the minimum cost clique cover problem in max-point tolerance graphs

A point-interval $$(I_v, p_v)$$ is a pair constituted by an interval $$I_v$$ of $${\mathbb {R}}$$ and a point $$p_v \in I_v$$ . A graph $$G=(V,E)$$ is a Max-Point-Tolerance (MPT) graph if each vertex $$v\in V$$ can be mapped to a point-interval in such a way that (u, v) is an edge of G iff $$I_u \cap I_v \supseteq \{p_u, p_v\}$$ . MPT graphs constitute a superclass of interval graphs and naturally arise in genetic analysis as a way to represent specific relationships among DNA fragments extracted from a population of individuals. One of the most important applications of MPT graphs concerns the search for an association between major human diseases and chromosome regions from patients that exhibit loss of heterozygosity events. This task can be formulated as a minimum cost clique cover problem in a MPT graph and gives rise to a $${{\mathcal {N}}}{{\mathcal {P}}}$$ -hard combinatorial optimization problem known in the literature as the Parsimonious Loss of Heterozygosity Problem (PLOHP). In this article, we investigate ways to speed up the best known exact solution algorithm for the PLOHP as well as techniques to enlarge the size of the instances that can be optimally solved. In particular, we present a Branch&Price algorithm for the PLOHP and we develop a number of preprocessing techniques and decomposition strategies to dramatically reduce the size of its instances. Computational experiments show that the proposed approach is 10–30 $$\times $$ faster than previous approaches described in the literature, and suggest new directions for the development of future exact solution approaches that may prove of fundamental assistance in practice.

Design of safe well on the top of Atharamura anticline, Tripura, India, on the basis of predicted pore pressure from seismic velocity data

Pore pressure prediction is an essential part of wildcat well planning. In India, Tripura sub-basin is characterised by huge anticlines, normal faults and abnormally pressured formations. These factors push the wildcat well planning in this area into wide margin of uncertainty. Pore pressures were predicted from seismic velocities by using modified Eaton’s method over the synclinal and flank part of Atharamura to understand the pressure succession towards the anticline. These predicted pore pressures on the flank part lead to a reasonable match when plotted with offset well-measured pore pressures. To reduce the uncertainty, fracture pressures were established by various methods such as Hubbert and Willis method and Matthews and Kelly method from predicted pore pressures. But the fracture pressures were predicted with available horizontal stress correlations due to lack of Poisson’s ratio curve for the study area. The mud pressure required to drill the well is calculated using median line principle, and hence drilling mud window is established by assuming virtual tight conditions. The plot of equivalent circulation density versus depth suggests that well can be drilled with two casing policy. But it is found that adding one more casing pipe will ensure the safety of well. Casing pipes were designed on the basis of collapse pressure, burst pressure and tensile load. Finally, a well plan which includes pore pressure, fracture pressure, drilling mud policy, casing policy and kick tolerance graph were proposed to give clear picture on well planning on the top of the anticline in pore pressure point of view.

A new approach to first tolerance evaluations in the conceptual design stage based on tolerance graphs

Tolerancing is commonly performed in late design stages, in which the product geometry is fully defined. However, this often leads to high manufacturing costs or expensive design changes. Therefore, motivated by the idea of frontloading, a new approach for first tolerance evaluations in conceptual design stages is developed, which enables a first tolerance specification and analysis based on the few geometry data available. Together with semantic information of the interface notations of adjacent parts, graphs including relevant geometry elements and tolerance information are generated. These graphs support designers with a first assessment of tolerance situations before the geometry is finally defined.

Intuitionistic fuzzy tolerance graphs with application

In this paper, intuitionistic fuzzy tolerance graphs and intuitionistic fuzzy $$\phi $$ -tolerance graphs are defined and investigated several interesting properties. Here $$\phi $$ is any real valued function. Particular cases of $$\phi $$ , viz. min, max and sum are considered. Also, intuitionistic fuzzy unit and proper tolerance graphs are defined and investigated many interesting results. A very appropriate application of intuitionistic fuzzy tolerance graph is also presented.

Fuzzy $$\phi $$ ϕ -tolerance competition graphs

In this paper, fuzzy \(\phi \)-tolerance competition graphs are defined. Several interesting properties are investigated. Here \(\phi \) is any real valued function. Particular cases of \(\phi \), viz. minimum, maximum and sum are considered. A daily life problem is modelled using fuzzy min-tolerance competition graph. Few properties are investigated for fuzzy tolerance graphs.

New Geometric Representations and Domination Problems on Tolerance and Multitolerance Graphs

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain amount of overlap without being in conflict. In one of the most natural generalizations of tolerance graphs with direct applications in the comparison of DNA sequences from different organisms, namely multitolerance graphs, two tolerances are allowed for each interval: one on the left side and the other on the right side. Several efficient algorithms for optimization problems that are NP-hard in general graphs have been designed for tolerance and multitolerance graphs. In spite of this progress, the complexity status of some fundamental algorithmic problems on tolerance and multitolerance graphs, such as the dominating set problem, remained unresolved until now---three decades after the introduction of tolerance graphs. In this paper we introduce two new geometric representations for tolerance and multitolerance graphs, given by points and line segments in the plane. Apart from being important on their own, these...

On recognition of threshold tolerance graphs and their complements

A graph G=(V,E) is a threshold tolerance graph if each vertex v∈V can be assigned a weight wv and a tolerance tv such that two vertices x,y∈V are adjacent if wx+wy≥min(tx,ty). Currently, the most efficient recognition algorithm for threshold tolerance graphs is the algorithm of Monma, Reed, and Trotter which has an O(n4) runtime. We give an O(n2) algorithm for recognizing threshold tolerance and their complements, the co-threshold tolerance (co-TT) graphs, resolving an open question of Golumbic, Weingarten, and Limouzy.

Generalized dynamic storage allocation

Graph Theory Dynamic Storage Allocation is a problem concerned with storing items that each have weight and time restrictions. Approximate algorithms have been constructed through online coloring of interval graphs. We present a generalization that uses online coloring of tolerance graphs. We utilize online-with-representation algorithms on tolerance graphs, which are online algorithms in which the corresponding tolerance representation of a vertex is also presented. We find linear bounds for the online-with-representation chromatic number of various classes of tolerance graphs and apply these results to a generalization of Dynamic Storage Allocation, giving us a polynomial time approximation algorithm with linear performance ratio.

On the intersection of tolerance and cocomparability graphs

Tolerance graphs have been extensively studied since their introduction, due to their interesting structure and their numerous applications, as they generalize both interval and permutation graphs in a natural way. It has been conjectured by Golumbic, Monma, and Trotter in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. Since cocomparability graphs can be efficiently recognized, a positive answer to this conjecture in the general case would enable us to efficiently distinguish between tolerance and bounded tolerance graphs, although it is NP-complete to recognize each of these classes of graphs separately. This longstanding conjecture has been proved under some– rather strong –structural assumptions on the input graph; in particular, it has been proved for complements of trees, and later extended to complements of bipartite graphs, and these are the only known results so far. Furthermore, it is known that the intersection of tolerance and cocomparability graphs is contained in the class of trapezoid graphs. Our main result in this article is that the above conjecture is true for every graph G that admits a tolerance representation with exactly one unbounded vertex; note that this assumption concerns only the given tolerance representation R of G, rather than any structural property of G. Moreover, our results imply as a corollary that the conjecture of Golumbic, Monma, and Trotter is true for every graph G=(V,E) that has no three independent vertices a,b,c∈V such that N(a)⊂N(b)⊂N(c), where N(v) denotes the set of neighbors of a vertex v∈V; this is satisfied in particular when G is the complement of a triangle-free graph (which also implies the above-mentioned correctness for complements of bipartite graphs). Our proofs are constructive, in the sense that, given a tolerance representation R of a graph G, we transform R into a bounded tolerance representation R∗ of G. Furthermore, we conjecture that any minimal tolerance graph G that is not a bounded tolerance graph, has a tolerance representation with exactly one unbounded vertex. Our results imply the non-trivial result that, in order to prove the conjecture of Golumbic, Monma, and Trotter, it suffices to prove our conjecture.

Counting independent sets in a tolerance graph

Counting independent sets is a #P-complete problem for general graphs but solvable in polynomial time for interval and permutation graphs. This paper develops some polynomial time algorithms for counting independent sets, maximal independent sets, and independent perfect dominating sets in a tolerance graph, which is a common generalization of interval and permutation graphs. No algorithm for solving those problems for tolerance graphs is currently available.