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Topic: Critical graph

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	Qrhetoric Calculus - Critical Points
		This is a point in the graph where on one side, the slop is increasing, and on the other side, the slope is decreasing.
		Critical points are the areas at which y' and y" are equal to 0, simply put.
		As you can see, the graph begins with a negative velocity, (displacement is decreasing) but it begins to slow its backward movement, which is reverse deceleration, or acceleration.
	www.qcalculus.com /cal06.htm (1667 words)

	Graph coloring - Wikipedia, the free encyclopedia
		In graph theory, graph coloring is an assignment of "colors", (red, blue and so on, but consecutive integers starting from 1 can be used without loss of generality), to certain objects in a graph.
		Graph coloring is not to be confused with graph labeling, which is an assignment of labels, usually also in the form of numbers, to vertices or edges.
		When used without any qualification, a coloring of a graph is always assumed to be a vertex coloring, namely an assignment of colors to the vertices of the graph.
	en.wikipedia.org /wiki/Graph_coloring (1580 words)

	How to Determine Critical Points - Free Math Help - Math Homework Help
		Critical points are the points on the graph where the derivative, or slope, equals zero or does not exist.
		It is often useful to find critical points because all relative maxima and minima occur at critical points.
		It is possible for a critical point to not be a relative extrema.
	www.freemathhelp.com /critical-points.html (320 words)

	Crystallographic Topology - Critical Nets 2
		Critical nets are actually Morse functions that are defined in terms of a mathematical mapping from Euclidean 3-space to Euclidean 1-space (i.e., a single valued 3-dimensional function).
		According to the special rhombohedral indexing, this point would have to be a degenerate critical point with a cubic (triple point) algebraic dependence rather than quadratic along the (e) to (f) vector since the density is heading downhill along that vector.
		Integer ratios of adjacent multiplicities provide the coordination vector, Coord, for the bcc critical graph (8,2,6,4,2,4) denoting 8 passes around a peak, 2 peaks around a pass, etc. as illustrated at the top of Fig.
	www.ornl.gov /sci/ortep/topology/critnet2.html (2301 words)

	Critical graph - Wikipedia, the free encyclopedia
		But in graph theory, when the term is used without any qualification, it almost always refers to the chromatic number of a graph.
		Critical graphs are interesting because they are the minimal members in terms of chromatic number, which is a very important measure in graph theory.
		A k-critical graph is a critical graph with chromatic number k; a graph G with chromatic number k is k-vertex-critical if each of its vertices is a critical element.
	en.wikipedia.org /wiki/Critical_graph (358 words)

	Gordon Royle's Small Graphs (Site not responding. Last check: 2007-10-20)
		If you are interested in small graph data that is not here, then feel free to mail me at gordon@cs.uwa.edu.au because I may have just not got around to installing it.
		The chromatic number of a graph is the smallest number k such that each vertex can be assigned a "colour" from the set {1,2,...,k} in such a fashion that the two ends of every edge are differently coloured.
		Vizing's theorem states that a graph can be edge-coloured in either d or d+1 colours, where d is the maximum degree of the graph.
	www.csse.uwa.edu.au /~gordon/graphs/index.html (708 words)

	Graph of a function - Wikipedia, the free encyclopedia
		In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching.
		For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem in functional analysis.
		The concept of the graph of a function is generalised to the graph of a relation.
	en.wikipedia.org /wiki/Graph_of_a_function (264 words)

	Topological techniques for shape understanding
		Generally the knowledge of the critical point configuration is considerated as one of the simplest way to describe the perception and the oraganization of the surface shape, [13].
		The classification of critical areas as maxima, minima and saddles is done by checking the number of non-constrained edges in the boundary and analysing the ascending/descending direction of the surface across the boundary, (for details in the classification see [10]).
		The concept of critical area is not able to fully describe the topologicl behaviour of the surface around saddles, so we have introduced the notion of influence zone of a critical area (see [10,11,23]).
	www.cescg.org /CESCG-2001/SBiasotti (3617 words)

	Gordon Royle's Small Graphs (Site not responding. Last check: 2007-10-20)
		If you are interested in small graph data that is not here, then feel free to mail me at gordon@cs.uwa.edu.au because I may have just not got around to installing it.
		The chromatic number of a graph is the smallest number k such that each vertex can be assigned a "colour" from the set {1,2,...,k} in such a fashion that the two ends of every edge are differently coloured.
		Vizing's theorem states that a graph can be edge-coloured in either d or d+1 colours, where d is the maximum degree of the graph.
	people.csse.uwa.edu.au /gordon/remote/graphs (708 words)

	Graph Coloring
		A graph is a collection of vertices (dots) and edges (lines) which connect the vertices.
		One portion of graph theory deals with coloring the vertices of a graph so that adjacent (joined) vertices are not colored the same.
		This graph is one of an infinite family of graphs which confirms a conjecture of G.A.Dirac from 1970.
	www.ma.iup.edu /cgi-bin/gallery?id=10 (203 words)

	AGG and CP-Analysis
		Critical pair analysis is known from term rewriting and usually used to check if a term rewriting system has a functional behavior, i.e.
		One rule application generates graph objects in a way that a graph structure would occur which is prohibited by a negative application condition (NAC) of another rule application.
		If the graph grammar analised is layered, the critical pair analysis can be optimized in the way that critical pairs are searched for rules in the same layer only.
	tfs.cs.tu-berlin.de /agg/critical_pairs.html (1451 words)

	New VHDL Based Critical Path Tracing Method for Fault Simulation by Massoud Shadfar, Armita Paymandoust, and ...
		After propagation of values corresponding to a test vector, all critical values will be determined by evaluation of the circuit in one pass from the output to the input of the circuit.
		Critical Path Graph is a graph formed by replacing logic gates with circles and fanouts with filled circles.
		of their input lines, the other decision made in one pass critical path tracing is the decision to mark a fault stem as being critical to not critical.
	www.ece.neu.edu /info/vhdl/test/cpt.html (5659 words)

	Abstracts of Discrete Math Seminars at SFU
		Abstract: Scheinerman and Wilfassert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph $K_n$." A "rectilinear drawing of $K_n$" is an arrangement of $n$ vertices in the plane, every pair of which is connected by an edge that is a line segment.
		Under this mapping, graphs, knots, finite groups and many infinite groups may be represented as bipartite graphs: in the case of finite groups and three connected graphs the reduction is complete.
		Let T be an even subset of vertices of graph G. A T-join is (the edge set of) any subgraph of G whose odd-degree vertices coincide with T. A T-cut is the set of edges having exactly one end in X, where X is any subset of vertices with X \cap T odd.
	www.math.sfu.ca /~goddyn/Seminars/abstracts.html (5920 words)

	Abstracts of Discrete Math Seminars at SFU (Site not responding. Last check: 2007-10-20)
		Abstract: Scheinerman and Wilfassert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph $K_n$." A "rectilinear drawing of $K_n$" is an arrangement of $n$ vertices in the plane, every pair of which is connected by an edge that is a line segment.
		Under this mapping, graphs, knots, finite groups and many infinite groups may be represented as bipartite graphs: in the case of finite groups and three connected graphs the reduction is complete.
		A topological obstruction for the torus is a graph G with minimum degree three that is not embeddable on the torus but for all edges e, G-e embeds on the torus.
	oldweb.cecm.sfu.ca /MRG/DMG/abstracts.html (9985 words)

	Math3343 Assignment 6, Fall 2002 (Site not responding. Last check: 2007-10-20)
		Lectures are vertices, two vertices are connected by an edge if they have a sstudent who wants to take both, that is, cannot be offered at the same time.
		Suppose that graph G is k-critical, and that G has a vertex v of degree less than k-1.
		19.8 (i) Let G be a toroidal graph drawn on the torus, with n vertices, f faces and m edges, which cannot be colored with 7 colors.
	www.cs.unb.ca /profs/horton/math3343/solution6.html (620 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		That is, if some PC was critical the previous time it was executed, we will predict that it is going to be critical the next _8_ times it is executed.
		Depending on what other paths are parallel to the critical path, a particular critical instruction may, in one extreme, have no paths nearby which are anywhere near as long, or it may, in the other extreme, have another nearby path which is just as long.
		The problem is that the graph is assigned fixed latencies from an execution trace, but the latency of load instruction may depend on the relative timing of other load instructions.
	moss.csc.ncsu.edu /~mueller/pact02/slides/tune.ppt (2009 words)

	New Page 1
		This main purpose for this change was to construct graphs which could be colored in classroom exercises.
		The output for this function is the 2-list whose first element is the upper bound on the chromatic number of the graph in question and whose second coordinate is the vertex partition so obtained.
		It was this particular partition and other ones similar to it that exhibited the pattern that led to a partial proof of the conjecture of Dirac.
	nsm1.nsm.iup.edu /jjl/GraphColoring/index.htm (882 words)

	Westgard QC Lesson: Critical-Error Graphs
		A critical-error graph is simply a power function graph that also displays the size of error that is medically important and needs to be detected by the QC procedure.
		With a critical-error graph, a quality planing model is used to determine the error condition of interest on the x-axis, then the power curve interpolated to determine the y-value or probability of detecting that error condition.
		The consequence is that the margin of safety may not be adequate if the budget has already been used up by the imprecision and inaccuracy of the method, thus you may be unhappy with the control rules and number of control measurements needed to adequately monitor the testing process.
	www.westgard.com /lesson5.htm (1212 words)

	Short term trading at Trade10.com- The science of trading market momentum for stocks, options, futures and bonds.
		Below is a graph of critical days on the SandP500 Index, and General Electric Co. As an index moves in short term price trends, generally, so do the bulk of stocks that compile to create the index value.
		Each critical day that we generate through our research is meant to indicate a point in the markets path when a reversal of the short trend has a higher probability of occurring.
		Often a failed critical day will indicate a stronger bias in the market for continuation of the trend that was in place prior to the critical day.
	www.trade10.com /short_term_trader.htm (751 words)

	Calculus I (Math 2413) - Applications of Derivatives - The Shape of a Graph, Part I
		may change signs is at the critical points of the function. We’ve now got another use for critical points. So, we’ll build a number line, graph the critical points and pick test points from each region to see if the derivative is positive or negative in each region.
		Once these points are graphed we go to the increasing and decreasing information and start sketching. For reference purposes here is the increasing/decreasing information.
		Note that we also know that the graph will be horizontal when it goes through each of the critical points. For each of the critical points the derivative was zero and so we know that the tangent line for each of these points must be horizontal.
	tutorial.math.lamar.edu /AllBrowsers/2413/ShapeofGraphPtI.asp (1436 words)

	[No title]
		Every graph G of maximum degree \Delta is (\Delta+1)-colourable and a classical theorem of Brooks states that G is not \Delta-colourable iff G has a (\Delta+1)-clique or \Delta = 2 and G has an odd cycle.
		We extend Reed's characterization of (\Delta-1)-colourable graphs and characterize (\Delta-2), (\Delta-3), (\Delta-4) and (\Delta-5)-colourable graphs, for sufficiently large \Delta, and prove a general structure for graphs with \chi close to \Delta.
		A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (k-1)-colourable.
	www.cs.mcgill.ca /~babak/paper.html (572 words)

	BAFL
		For example, by representing micro-execution events and dependences as a suitable dependence graph, its critical path automatically determines which processor stage (e.g., fetch, execute, or commit) is a bottleneck, and also for which dynamic instructions.
		Finally, observing that past criticality of an instruction correlates with its future criticality, we turned the critical-path analyzer into a criticality predictor, thus facilitating the design of first truly cost-aware processor policies.
		To exploit it, we developed a hardware slack analyzer (thanks to a reduction trick, as simple the criticality analyzer) and used it to systematically devise a control policy for a power-friendly processor in which one half is clocked at half the frequency.
	www.cs.berkeley.edu /~bfields/bafl (912 words)

	PIGALE Library
		It is proved that any plane graph may be represented by a triangle contact system, that is a collection of triangular disks which are disjoint except at contact points, each contact point being a node of exactly one triangle.
		The edges of any maximal bipartite plane graph G with outer face bwb'w' can be colored by two colors such that the color classes form spanning trees of G-b and G-b', respectively.
		We give a characterization of DFS cotree-critical graphs which is central to the linear time Kuratowski finding algorithm implemented in PIGALE (Public Implementation of a Graph Algorithm Library and Editor) by the authors, and deduce an algorithm for finding a Kuratowski subdivision in a DFS cotree-critical graph.
	pigale.sourceforge.net /bibliography.html (804 words)

	Wayne Hayes's Ramsey Graph Research (Site not responding. Last check: 2007-10-20)
		For my course project in Derek Corneil's graduate Graph Theory course CSC 2410, I decided to try throwing some serious brute force at finding graphs which would put lower bounds on small Ramsey Numbers.
		This circulant is a critical graph because R(3,9) is known to be equal to 36.
		We found another critical circulant on 24 nodes for R(4,5), which is known to be 25.
	www.cs.toronto.edu /~wayne/research/ramsey/ramsey.html (436 words)

	aiSee Graph of the Month 10/01: Dhrystone Call Graph
		Call graph of the Dhrystone benchmark application for the C16x/ST10 family of microcontrollers.
		Stack height differences can be shown as annotations in the call graph and control flow graph.
		Critical program sections can then be easily recognized thanks to color coding.
	www.aisee.com /graph_of_the_month/aicall.htm (265 words)

	Petr Hlineny: Scientific Research
		Informally, the vertices of the intersection graph are the sets from M, and the edges of G connect intersecting pairs of these sets.
		The crossing number of a graph G, denoted by cr(G), is defined as the smallest possible number of edge-crossings in a drawing of G in the plane.
		Unlike graphs, matroids are very hard to imagine (or to draw a picture) and to apply even simple operations to them.
	www.fi.muni.cz /~hlineny/research.html (1516 words)

	Implementation
		Given an activity-node graph, the objective of critical path analysis is to determine the slack time for each activity and thereby to identify the critical activities and the critical path.
		In the implementation shown, an edge-weighted graph is constructed that is isomorphic with the the original event-node graph, but in which the edge weights are the slack times as given by Equation
		By constructing such a graph we can make use of Dijkstra's algorithm find the shortest path from start to finish since the shortest path must be the critical path (line 34).
	www.brpreiss.com /books/opus4/html/page583.html (388 words)

	Using VHDL Critical Path Tracing Models for Pseudo Random Test Generation - Massoud Shadfar, Armita Paymandoust, ...
		Critical path tracing is used for finding faults detected by a specific test vector.
		The argument used in conjunction with Figure 4 for finding critical path lines can be used to mark all the critical lines of Figure 5 except the fanout stem.
		Above all, the case mentioned before, in which a fanout stem can be critical even if none of its branches are critical, cannot be handled by the standard CPT method.
	www.ece.neu.edu /info/vhdl/test/ran_test.html (5576 words)

	AMCA: Construction of Cores by Zhongyuan Che (Site not responding. Last check: 2007-10-20)
		A graph homomorphism from G to H is a mapping f:V(G) --> V(H) such that g
		A graph G is chi-critical if the chromatic number of every proper subgraph are strictly less than chi(G).
		A (k, 1)-coloring of G is simply a proper k-coloring of G; therefore the chromatic number chi(G) is the smallest k for which G admits a (k, 1)-coloring.
	at.yorku.ca /c/a/g/s/50.htm (227 words)

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