scholarly journals Recognition and Characterization of Chronological Interval Digraphs

10.37236/2497 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Sandip Das ◽  
Mathew Francis ◽  
Pavol Hell ◽  
Jing Huang
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Recognition Algorithm ◽  
Interval Graphs ◽  
Structure Characterization ◽  
Left Endpoint ◽  
New Class ◽  
Closed Intervals ◽  
Novel Structure ◽  
Definition Of

Interval graphs admit elegant structural characterizations and linear time recognition algorithms; on the other hand, the usual interval digraphs lack a forbidden structure characterization as well as a low-degree polynomial time recognition algorithm. In this paper we identify another natural digraph analogue of interval graphs that we call ”chronological interval digraphs”. By contrast, the new class admits both a forbidden structure characterization and a linear time recognition algorithm. Chronological interval digraphs arise by interpreting the standard definition of an interval graph with a natural orientation of its edges. Specifically, $G$ is a chronological interval digraph if there exists a family of closed intervals $I_v$, $v \in V(G)$, such that $uv$ is an arc of $G$ if and only if $I_u$ intersects $I_v$ and the left endpoint of $I_u$ is not greater than the left endpoint of $I_v$. (Equivalently, if and only if $I_u$ contains the left endpoint of $I_v$.)We characterize chronological interval digraphs in terms of vertex orderings, in terms of forbidden substructures, and in terms of a novel structure of so-called $Q$-paths. The first two characterizations exhibit strong similarity with the corresponding characterizations of interval graphs. The last characterization leads to a linear time recognition algorithm.

scholarly journals Closed graphs are proper interval graphs

2014 ◽  
Vol 22 (3) ◽  
pp. 37-44
Author(s):  
Marilena Crupi ◽  
Giancarlo Rinaldo
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Simple Graph ◽  
Interval Graphs ◽  
Closed Graph ◽  
Graph Recognition ◽  
Proper Interval Graph ◽  
Linear Time Algorithms

Abstract Let G be a connected simple graph. We prove that G is a closed graph if and only if G is a proper interval graph. As a consequence we obtain that there exist linear-time algorithms for closed graph recognition.

scholarly journals A four-sweep LBFS recognition algorithm for interval graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Peng Li ◽  
Yaokun Wu
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Recognition Algorithm ◽  
Structure Theory ◽  
Time Interval ◽  
Breadth First Search ◽  
Input Graph ◽  
Graph Recognition ◽  
Unit Interval Graph ◽  
International Audience

Graph Theory International audience In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic Breadth-First Search (LBFS) and prove its correctness. They believe that their corresponding 5-sweep LBFS interval graph recognition algorithm is also correct. Thanks to the LBFS structure theory established mainly by Corneil et al., we are able to present a 4-sweep LBFS algorithm which determines whether or not the input graph is a unit interval graph or an interval graph. Like the algorithm of Corneil et al., our algorithm does not involve any complicated data structure and can be executed in linear time.

scholarly journals Linear Time Recognition Algorithms and Structure Theorems for Bipartite Tolerance Graphs and Bipartite Probe Interval Graphs

2010 ◽  
Vol Vol. 12 no. 5 (Graph and Algorithms) ◽  
Author(s):  
David E. Brown ◽  
Arthur H. Busch ◽  
Garth Isaak
Keyword(s):  
Real Line ◽  
Positive Real Number ◽  
Linear Time ◽  
Interval Graph ◽  
Interval Graphs ◽  
Positive Real ◽  
Recognition Algorithms ◽  
Probe Interval Graphs ◽  
Tolerance Graphs ◽  
International Audience

Graphs and Algorithms International audience A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes with an interval associated to each vertex so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is a probe. A graph G = (V, E) is a tolerance graph if each vertex v is an element of V can be associated to an interval I(v) of the real line and a positive real number t(v) such that uv is an element of E if and only if vertical bar I(u) boolean AND I(v)vertical bar >= min \t(u), t(v)\. In this paper we present O(vertical bar V vertical bar + vertical bar E vertical bar) recognition algorithms for both bipartite probe interval graphs and bipartite tolerance graphs. We also give a new structural characterization for each class which follows from the algorithms.

scholarly journals Minimal Obstructions for Partial Representations of Interval Graphs

10.37236/5862 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Pavel Klavik ◽  
Maria Saumell
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Interval Graphs ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Intersection Graphs ◽  
Certifying Algorithm ◽  
Partial Representation ◽  
Partial Representation Extension ◽  
Linear Time Algorithms

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals.  We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension.

A Real Time Control Architecture for Continuously Managing Patients in a Care Unit

1995 ◽  
Vol 34 (05) ◽  
pp. 475-488
Author(s):  
B. Seroussi ◽  
J. F. Boisvieux ◽  
V. Morice
Keyword(s):  
Real Time ◽  
Linear Time ◽  
Treatment Plan ◽  
Control Architecture ◽  
Real Time Control ◽  
Time Representation ◽  
Time Control ◽  
Continuous Support ◽  
Definition Of ◽  
Computerized System

Abstract:The monitoring and treatment of patients in a care unit is a complex task in which even the most experienced clinicians can make errors. A hemato-oncology department in which patients undergo chemotherapy asked for a computerized system able to provide intelligent and continuous support in this task. One issue in building such a system is the definition of a control architecture able to manage, in real time, a treatment plan containing prescriptions and protocols in which temporal constraints are expressed in various ways, that is, which supervises the treatment, including controlling the timely execution of prescriptions and suggesting modifications to the plan according to the patient’s evolving condition. The system to solve these issues, called SEPIA, has to manage the dynamic, processes involved in patient care. Its role is to generate, in real time, commands for the patient’s care (execution of tests, administration of drugs) from a plan, and to monitor the patient’s state so that it may propose actions updating the plan. The necessity of an explicit time representation is shown. We propose using a linear time structure towards the past, with precise and absolute dates, open towards the future, and with imprecise and relative dates. Temporal relative scales are introduced to facilitate knowledge representation and access.

scholarly journals Some Inequalities for LR-$$\left({h}_{1}, {h}_{2}\right)$$-Convex Interval-Valued Functions by Means of Pseudo Order Relation

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Kottakkaran Sooppy Nisar ◽  
Khadiga Ahmed Ismail ◽  
...  
Keyword(s):  
Applied Mathematics ◽  
Critical Role ◽  
Order Relation ◽  
Strong Relationship ◽  
New Class ◽  
Starting Point ◽  
Interval Valued Function ◽  
Fejér Inequality ◽  
Definition Of ◽  
Interval Valued

AbstractIn both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and symmetry is strong. Whichever one we work on, we introduced the new class of generalized convex function is known as LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued function (LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -IVF) by means of pseudo order relation. Then, we established its strong relationship between Hermite–Hadamard inequality (HH-inequality)) and their variant forms. Besides, we derive the Hermite–Hadamard–Fejér inequality (HH–Fejér inequality)) for LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued functions. Several exceptional cases are also obtained which can be viewed as its applications of this new concept of convexity. Useful examples are given that verify the validity of the theory established in this research. This paper’s concepts and techniques may be the starting point for further research in this field.

Deterministic Continuous-Time Signals and Systems

2021 ◽  
pp. 562-598
Author(s):  
Stevan Berber
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Linear Time ◽  
Continuous Variable ◽  
Linear Time Invariant ◽  
Time Signals ◽  
Analysis And Synthesis ◽  
Linear Time Invariant Systems ◽  
Definition Of ◽  
Time Invariant

Due to the importance of the concept of independent variable modification, the definition of linear-time-invariant system, and their implications for discrete-time signal processing, Chapter 11 presents basic deterministic continuous-time signals and systems. These signals, expressed in the form of functions and functionals such as the Dirac delta function, are used throughout the book for deterministic and stochastic signal analysis, in both the continuous-time and the discrete-time domains. The definition of the autocorrelation function, and an explanation of the convolution procedure in linear-time-invariant systems, are presented in detail, due to their importance in communication systems analysis and synthesis. A linear modification of the independent continuous variable is presented for specific cases, like time shift, time reversal, and time and amplitude scaling.

scholarly journals Towards a Theory for Unintended Consequences in Engineering Design

2019 ◽  
Vol 1 (1) ◽  
pp. 3411-3420
Author(s):  
Hannah Walsh ◽  
Andy Dong ◽  
Irem Tumer
Keyword(s):  
Failure Analysis ◽  
Engineering Design ◽  
Bifurcation Analysis ◽  
Real World ◽  
Unintended Consequences ◽  
Novel Technologies ◽  
New Class ◽  
Failure State ◽  
Engineered Systems ◽  
Definition Of

AbstractConventional failure analysis ignores a growing challenge in the responsible implementation of novel technologies into engineered systems - unintended consequences, which impact the engineered system itself and other systems including social and environmental systems. In this paper, a theory for unintended consequences is developed. The paper proposes a new definition of unintended consequences as behaviors that are not intentionally designed-into an engineered system yet occur even when a system is operating nominally, that is, not in a failure state as conventionally understood. It is argued that the primary cause for this difference is the bounded rationality of human designers. The formation of unintended consequences is modeled with system dynamics, using a specific real-world example, and bifurcation analysis. The paper develops propositions to guide research in the development of new design methods that could mitigate or control the occurrence and impact of unintended consequences. The end goal of the research is to create a new class of failure analysis tools to manage unintended consequences responsibly to facilitate engineering design for a more sustainable future.

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