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Topic: Interval graph

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	Graph Piecewise Functions
		In the interval [2, + inf) the graph is a line with an x intercept at (3, 0) and passes through the point (2, 1).
		In the interval (- inf, 0) the graph of f is a hyperbola with vertical asymptote at x = 0.
		In the interval [0, + inf) the graph is a decreasing exponential and passes through the point (0, 1).
	www.analyzemath.com /Graphing/piecewise_functions.html (920 words)

	Interval graph - Wikipedia, the free encyclopedia
		In graph theory, an interval graph is a graph that captures the intersections among a set of intervals on the real line.
		Interval graphs are useful in modeling resource allocation problems in operations research.
		Interval graphs are chordal graphs and hence perfect graphs.
	en.wikipedia.org /wiki/Interval_graph (215 words)

	DCI 2001 Research Program Abstracts - Week 1
		Interval graphs and probe interval graphs were introduced for studies in certain biological fields specializing in genetics in the late 50's (for interval graphs), and in the late 90's (for probe interval graphs).
		A probe interval graph is an variation of interval graph that arose from the DNA physical mapping of molecular biology.
		Unit probe interval graph is a special case of probe interval graph where we require that all the intervals assigned to the vertices must have the same length.
	dimacs.rutgers.edu /dci/2001/abstractswk1right.html (2607 words)

	Help - www.eniro.com
		To zoom in a shorter time interval, you can click and drag the cursor horizontally alongside the graph.
		To reset the graph, use the time interval menu or state your own dates below the graph.
		Click on the boxes and the index graph line will be displayed in the same color.
	www.eniro.com /en/Special-Pages/Help (624 words)

	CU-Denver Department of Mathematics
		A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes and intervals can be assigned to vertices so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of the vertices is a probe.
		Interval point bigraphs (and the probe interval graphs to which they correspond) are characterized by properties of their reduced adjacency matrices, as are unit interval bigraphs and interval bigraphs.
		A circular arc graph is the intersection graph of arcs of a circle.
	www-math.cudenver.edu /events/these.shtml (1624 words)

	[No title]
		A graph is called an interval graph if it is isomorphic to the intersection graph of a collection of closed intervals on the real line.
		Interval graphs arose in the study of genetics as a way of attempting to test whether overlap data about gene fragments was consistent with a linear arrangement of the components of genes.
		Interval orders arise in scheduling problems and the special case of unit interval orders, aka semiorders (having a representation in which all intervals have the same length) arises in mathematical psychology and in statistics.
	www.socsci.uci.edu /imbs/Workshops/OSDA/BOGART.htm (259 words)

	[No title]
		Interval graphs are characterised as those triangulated graphs which do not possess an asteroidal triple.
		Infinte interval graphs are characterized by a new form of decomposition generalizing consecutive simplicial decomposition.
		A graph $G$ is an interval graph if and only if every quadrilateral in $G$ has a diagonal and every odd cycle in $G^{c}$ has a triangular chord.
	www.cs.ualberta.ca /~stewart/GRAPH/interval/interval.bib (655 words)

	5.1 Graph-based Interval Representation (Site not responding. Last check: 2007-10-26)
		Given a graph where a precedes some point n that precedes b then the addition of an edge between a and b will be redundant (see Figure 5.1ii).
		Given a graph where points a and b are both preceded by a third point n, adding an edge between a and b will cause one of the edges to be redundant.
		The construction time of a graph is dependent upon two factors: the time taken to verify if the relation is already represented in the graph (constant time), and the time taken to search if the inverse relation is true (one search).
	www.computing.edu.au /~jc/travt/thesis/node27.html (1722 words)

	Interval Bigraphs (Site not responding. Last check: 2007-10-26)
		There is a notion similiar to that of asteroidal triples in graphs: An asteroidal triple of edges is formed by three edges where any two are connected by a path avoiding vertices and neighbors of the third edge.
		Therefore, recognizing interval bigraphs is at least as difficult than recognizing interval graphs.
		The difficulty of recognizing interval bigraphs may be due to the fact that, although intervals of the real line have the Helly-property, the bicliques, i.e.
	www.math.uni-hamburg.de /spag/gd/mitarbeiter/prisner/Pris/ExBiInterval.html (531 words)

	Evergreens (Site not responding. Last check: 2007-10-26)
		If the knowledge graph of the students is connected, and if there are at least 4 students, then we may find out all attendence lists in polynomial time, and these lists are unique (modulo relabeling of the courses---of course we don't know the names of the courses).
		The classes of line graphs and interval graphs are both algorithmically relatively tame: Membership in both classes can be recognized in polynomial time, furthermore many optimization problems that are NP-hard for general graphs are polynomial-time solvable when restricting the input to interval or line graphs.
		This, and the prominence of interval graphs and line graphs seduced some people to take the opinion that problems for intersection graphs should always be relatively easy to tackle.
	www.math.uni-hamburg.de /spag/gd/mitarbeiter/prisner/Pris/Evergreens.html (562 words)

	CU-Denver Department of Mathematics Events
		We give necessary and sufficient conditions for an interval k-graph to be the complement of a comparability (transitively orientable) graph, in terms of the interval representation and discuss progress towards a characterization in terms of forbidden substructures.
		Abstract: An interval graph is a graph whose vertices are labeled with intervals on the real line, with an edge between vertices if and only if their corresponding intervals overlap.
		Abstract: An interval bigraph is the bipartite intersection graph of two distinct families of intervals in which vertices are adjacent if and only if their corresponding intervals overlap and each belongs to a distinct class of intervals.
	www-math.cudenver.edu /events/QueryEvent.php?eid=59 (690 words)

	BasicGraphs.htm (Site not responding. Last check: 2007-10-26)
		The graphs of all of these functions extend to infinity in at least one direction and it is part of the study of calculus to understand what these graphs do as they leave the small window in which we are graphing them.
		Note that the graph of g is the graph of f translated up 3 units and 4 units to the left.
		Both graphs are increasing for all values in the domain.
	www.utc.edu /Faculty/Terry-Walters/151/BasicGraphs.htm (1070 words)

	CS 762 (Graph-Theoretic Algorithms), Fall 2005: Lecture Summaries
		Not all graphs are interval graphs, in particular, an interval graph may not contain an induced cycle of length >= 4.
		A graph is an interval graph if and only if its maximal cliques can be ordered such that for any vertex, the cliques containing it are consecutive.
		Graphs of bounded pathwidth: same as treewidth, except that the tree must be a path.
	www.student.cs.uwaterloo.ca /~cs762/Summaries/index.php (2348 words)

	Web crawler - Wikipedia, the free encyclopedia
		A partial solution to these problems is the robots exclusion protocol, also known as the robots.txt protocol (Koster, 1996) that is a standard for administrators to indicate which parts of their Web servers should not be accessed by robots.
		This standard does not include a suggestion for the interval of visits to the same server, even though this interval is the most effective way of avoiding server overload.
		The first proposal for the interval between connections was given in (Koster, 1993) and was 60 seconds.
	en.wikipedia.org /wiki/Web_crawler (4590 words)

	chordalExperiments.html (Site not responding. Last check: 2007-10-26)
		The interval coloring problem models the compile-time memory allocation problem and has a rich history dating back at least to the 1970's.
		Both problems are NP-complete even for interval graphs, though there are constant-factor approximation algorithms for both problems on interval graphs.
		Our overall conclusion is that first-fit is a practical, easy to implement heuristic, yielding close to optimal performance, even for applications in which the underlying graph is more complicated than an interval graph.
	www.cs.uiowa.edu /~rraman/chordalGraphExperiments.html (481 words)

	SuperMemo: Learning statistics dialog
		For that reason, the initial value of the RF matrix is taken from the model of a less-than-average student (the model of average student is not used because the convergence from poorer student parameters upwards is faster than the convergence in the opposite direction).
		This graph is used to compute the estimated forgetting index that in turn is used to normalize grades (for delayed or advanced repetitions) and estimate the new value of item's A-Factor.
		To zoom in on a portion of the graph (as in pictures below), sweep the portions of the graphs that are to be removed with the mouse (e.g.
	www.supermemo.com /help/analysis.htm (1674 words)

	SuperMemo: Learning statistics dialog (Site not responding. Last check: 2007-10-26)
		These correspond to twenty repetition number categories multiplied by twenty A-Factor categories (note that for data representation convenience, the columns of the RF matrix for the first repetition are indexed by the number of memory lapses rather than A-Factor).
		This graph is used by the Algorithm SM-8 to quickly estimate the first value of A-Factor at the moment when all we know about an element is the first grade it has scored in its first repetition
		The first interval graph shows exponential regression curve that approximates the length of the first interval for different numbers of memory lapses (including the zero-lapses category that corresponds with newly memorized items).
	www.supermemo.com /archive/help2000/analysis.htm (1343 words)

	Steven Mills
		The feature interval graph is recursively computed, making it a compact representation, and uses an interval model of uncertainty.
		The methods presented are based on a graph structure, the feature interval graph, which is constructed from information provided by low-level processes such as stereo and motion tracking.
		Interval arithmetic is a method for performing computations on measurements that are only known to within a fixed error range.
	www.cs.nott.ac.uk /~smx/publications.html (1413 words)

	[No title]
		Every circle graph is $2 \omega$ colourable, and such a colouring can be found in $O(n log n)$ time.
		It's also shown that a graph $G$ is a permutation graph iff both $G$ and its complement are transitively orientable(i.e.
		In particular, comparability, cocomparability and permutation graphs are considered.", volume= 90, year= 1992, pages= "33-55") @article(CoPe84, author= "D.G. Corneil and Y. Pearl", title= "Clustering and domination in perfect graphs", journal= "Discrete Applied Mathematics", annote= "The $k$-cluster and the $k$-domination problems are known to be {NP}-complete for graphs in general.
	www.cs.ualberta.ca /~stewart/GRAPH/search/all.bib (1767 words)

	5 Primitive Interval Representations where some relations are unknown (Site not responding. Last check: 2007-10-26)
		In this chapter we expand the point-based system of representing intervals to encompass the types of interval datasets where information about every pair of intervals is not explicitly known but whenever interval relations are known they are only primitive interval relations.
		A variety of different types of graphs are randomly generated and a suite of searches performed on each of these graphs.
		A discussion of the relevance of the interval representation presented in this chapter is given in Section 5.4 and the conclusions in Section 5.5.
	www.computing.edu.au /~jc/travt/thesis/node26.html (352 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		4.) Triangulated graphs* as intersection graphs (Golumbic Ch.
		Tolerance graphs: * constant tolerance graphs are interval graphs * tolerance graphs with tolerance=interval length are permutation graphs.
		Comparability Graphs: June 09 9 * Recognition algorithm and complexity * perfectness and Dilworth theorem.
	www.math.tau.ac.il /~rshamir/atga/archive/outline.94 (436 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		It is an extension of the interval graph.
		Probe interval graph arises in the physical mapping contexts of human genome project.
		STS interval graph (STSIG) is a special case of PIG with all the p-vertices are not overlapping.
	dimacs.rutgers.edu /Workshops/DNAMapping/abstracts/dna33.html (132 words)

	CU CSCI Colloquium - McConnell (2001-2002)
		Given a set of intervals on a line, an interval graph is derived by creating a vertex for each interval and installing an edge for each intersecting pair of intervals.
		Interval-graph recognition is the inverse of this operation, namely, deriving a corresponding set of intervals given only the interval graph.
		Circular-arc graph and circular-arc graph recognition are defined similarly, except that they deal with intersections of arcs on a circle.
	www.cs.colorado.edu /events/colloquia/2001-2002/mcconnell.html (228 words)

	[No title]
		Explain why the sum of the degrees of the points of a graph equals twice the number of edges.
		Define a dual graph of a planar graph.
		Show that every planar graph with 11 points has at least one point vertex of degree at most four.
	www.cbu.edu /~yanushka/m405/v.1 (693 words)

	ADM Seminar
		In an interval graph, each vertex is associated with an interval on the real line with two vertices adjacent if and only if their associated intervals intersect.
		In 1982 Golumbic and others suggested associating "tolerances" with each interval so that now two vertices are joined if and only if the length of the intersection of their associated intervals is at least the minimum of the two tolerances.
		In the investigation of phi-tolerance interval graphs, a particular case appeared to be of recurring interest ---namely, when all the intervals are nested to form a chain under inclusion.
	www.math.clemson.edu /~gmatthe/admF02.html (1818 words)

	[No title]
		Then, * I assign each interval into a set, depending on what kind of a subgraph this interval * forms with the other intervals it overlaps with.
		Such subgraph is formed * when all the intervals that the current subgraph overlaps with also overlap with * each other.
		It returns * a negative value if the beginning point of the interval that calls the function * is smaller than that of the interval passed in as a parameter.
	www-cse.ucsd.edu /~ytsipeny/home/final_project/Optimal.java (516 words)

	p-centers and p-medians in interval and circular-arc graphs (Site not responding. Last check: 2007-10-26)
		Efficient algorithms for centers and medians in interval and circular-arc graphs
		We provide, given the interval model of an n vertex interval graph, an O(n) time algorithm for the 1-median problem on the interval graph.
		We introduce a spring model of computation and show how to solve the p-center problem on an circular-arc graph in O(pn) time, assuming that the arc endpoints are sorted.
	www.cs.ubc.ca /spider/besp/medians.html (215 words)

	SIDMA Volume 10 Issue 4
		This problem arises in numerous applications in which topological information on intersection of pairs of intervals is accompanied by additional metric information on their order, distance, or size.
		Our results are (1) a polynomial algorithm for the problem on interval graphs which admit a unique clique order (UCO graphs).
		(2) In case all constraints are upper and lower bounds on individual interval lengths, the problem on UCO graphs is linearly equivalent to deciding if a system of difference inequalities is feasible.
	epubs.siam.org /sam-bin/dbq/article/30637 (249 words)

	PACT-97 Article: L. Chen. Tight Lower Bounds for Computing Shortest Paths on Proper Interval and Bipartite Permutation ... (Site not responding. Last check: 2007-10-26)
		Logarithmic time lower bounds for computing the distance between two arbitrary vertices, in a proper interval graph represented by a family of intervals on a real line, and in a bipartite permutation graph represented by a permutation function, on exclusive write PRAM are proved here.
		The lower bounds are also valid for these classes of graphs represented by adjacency matrices and for their superclasses.
		Shortest paths on interval and permutation graphs, which, respectively, strictly contain proper interval and bipartite permutation graphs, are known to be computable in logarithmic time on exclusive write PRAM.
	www.pact.sscc.ru /conference/pact97/articles/6661277002.html (167 words)

	Chapter 2.6 (Site not responding. Last check: 2007-10-26)
		Note that the interval notation is always read left to right. So the first “number” we begin with here is negative infinity…and we stop at –4. Thus we get, in interval notation:
		Reading the interval notation needs to be as natural as interpreting a number line. When you are given the interval notation, you should be able to “picture” the solution in you head (i.e.
		I will leave the graph and interval notation to you.
	www.mtsu.edu /~smcdanie/CSS_Site/Fall2003/Chapter2/Section2_6/Section2_6.htm (601 words)

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