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From Wikipedia

Circle graph

In graph theory, a circle graph is a graph whose vertices can be associated with chords of a circle such that two vertices are adjacent if and only if the corresponding chords in the circle intersect. In other words, a circle graph is the intersection graph of a set of chords of a circle.

Algorithmic complexity

gives an O(n2)-time recognition algorithm for circle graphs that also computes a circle model of the input graph if it is a circle graph.

A number of other problems that are NP-complete on general graphs have polynomial time algorithms when restricted to circle graphs. For instance, showed that the treewidth of a circle graph can be determined, and an optimal tree decomposition constructed, in O(n3) time. Additionally, a minimum fill-in (that is, a chordal graph with as few edges as possible that contains the given circle graph as a subgraph) may be found in O(n3) time. has shown that a maximum clique of a circle graph can be found in O(nlog2n) time, while have shown that a maximum independent set of an unweighted circle graph can be found in O(nmin{d, α}) time, where d is a parameter of the graph known as its density, and α is the independence number of the circle graph.

However, there are also problems that remain NP-complete when restricted to circle graphs. These include the minimum dominating set, minimum connected dominating set, and minimum total dominating set problems.

Chromatic number

The chromatic number of a circle graph is the minimum number of colors that can be used to color its chords so that no two crossing chords have the same color. Since it is possible to form circle graphs in which arbitrarily large sets of chords all cross each other, the chromatic number of a circle graph may be arbitrarily large, and determining the chromatic number of a circle graph is NP-complete. However, several authors have investigated problems of coloring restricted subclasses of circle graphs with few colors. In particular, for circle graphs in which no sets of k or more chords all cross each other, it is possible to color the graph with as few as 2k + 6 colors. In the particular case when k = 3 (that is, for triangle-free circle graphs) the chromatic number is at most five, and this is tight: all triangle-free circle graphs may be colored with five colors, and there exist triangle-free circle graphs that require five colors. If a circle graph has girth at least five (that is, it is triangle-free and has no four-vertex cycles) it can be colored with at most three colors.

Applications

Circle graphs arise in VLSIphysical design as an abstract representation for a special case for wire routing, known as "two-terminal switchbox routing". In this case the routing area is a rectangle, all nets are two-terminal, and the terminals are placed on the perimeter of the rectangle. It is easily seen that the intersection graph of these nets is a circle graph. Among the goals of wire routing step is to ensure that different nets stay electrically disconnected, and their potential intersecting parts must be laid out in different conducting layers. Therefore circle graphs capture various aspects of this routing problem.

Colorings of circle graphs may also be used to find book embeddings of arbitrary graphs: if the vertices of a given graph G are arranged on a circle, with the edges of G forming chords of the circle, then the intersection graph of these chords is a circle graph and colorings of this circle graph are equivalent to book embeddings that respect the given circular layout.

Related graph classes

A graph is a circle graph if and only if it is the overlap graph of a set of intervals on a line. This is a graph in which the vertices correspond to the intervals, and two vertices are connected by an edge if the two intervals overlap, with neither containing the other.

The intersection graph of a set of intervals on a line is called the interval graph.

String graphs, the intersection graphs of curves in the plane, include circle graphs as a special case.

Every distance-hereditary graph is a circle graph, as is every permutation graph. Every outerplanar graph is also a circle graph.



From Encyclopedia

Graphs Graphs

A graph is a pictorial representation of the relationship between two quantities. A graph can be anything from a simple bar graph that displays the measurements of various objects to a more complicated graph of functions in two or three dimensions. The former shows the relationship between the kind of object and its quantity; the latter shows the relationship between input and output. Graphing is a way to make information easier for a viewer to absorb. The simplest graphs show the number of many objects. For example, a bar graph might name the months of the year along a horizontal axis and show numbers (for the number of days in each month) along a vertical axis. Then a rectangle (or bar) is drawn above each month. The height of the bar might indicate the number of days in that month on which it rained, or on which a person exercised, or on which the temperature rose above 90 degrees. See the generic example of a bar graph below (top right). Another simple kind of graph is a circle graph or pie graph, which shows fractions or percentages. In this kind of graph, a circle is divided into pieshaped sectors. Each sector is given a label and indicates the fraction of the total area that goes with that label. See the generic example of a pie graph below (top middle). A pie chart might be used to display the percentages of a budget that are allotted to various expenditures. If the sector labeled "medical bills" takes up two-tenths of the area of the circle, that means that two-tenths, or 20 percent, of the budget is devoted to medical expenses. Usually the percentages are written on the sectors along with their labels to make the graph easier to read. In both bar graphs and pie graphs, the reader can immediately pick out the largest and smallest categories without having to search through a chart or list, making it easy to compare the relative sizes of many objects simultaneously. Often the two quantities being graphed can both be represented numerically. For example, student scores on examinations are often plotted on a graph, especially if there are many students taking the exam. In such a graph, the numbers on the horizontal axis represent the possible scores on the exam, and the numbers on the vertical axis represent numbers of students who earned that score. The information could be plotted as a simple bar graph. If only the top point of each bar is plotted and a curve is drawn to connect these points, the result is a line graph. See the generic example of a line graph on the pevious page (top left). Although the points on the line between the plotted points do not correspond to any pieces of information, a smooth line can be easier to understand than a large collection of bars or dots. Graphs become slightly more complicated when one (or both) of the quantities in the graph can have continuous values rather than a discrete set. A common example of this is a quantity that changes over time. For example, a scientist might be observing the rate of growth of bacteria. The rates could be plotted so that the horizontal axis displays units of time and the vertical axis displays numbers (how many bacteria exist). Then, for instance, the point (3,1000) would mean that at "time 3" (which could mean three o'clock, or three seconds after starting, or various other times, depending on the units being used and the starting point of the experiment) there were one thousand bacteria in the sample. The rise and fall of the graph show the increases and decreases in the number of bacteria. In this case, even though only a finite set of points represent actual data, the remaining points do have a natural interpretation. For instance, suppose that in addition to the point (3,1000), the graph also contains the point (4,1500) and that both of these points correspond to actual measurements. If the scientist joins all of the points on the graph by a line, then the point (3.5,1200) might lie on the graph, or perhaps the point (3.5,1350). There are many different lines that can be drawn through a collection of points. Looking at the overall shape of the data points helps the scientist decide which line is the most reasonable fit. In the previous example, the scientist could estimate that at time 3.5, there were 1200 (or 1350) bacteria in the sample. Thus graphing can be helpful in making estimates and predictions. Sometimes the purpose for drawing a graph may not be to view the data already known but to construct a mathematical model that will allow one to analyze data and make predictions. One of the simplest models that can be constructed from a set of data is called a best-fit line. Such a line is useful in situations in which the data are roughly linear—that is, they are increasing or decreasing at a roughly constant rate but do not fall precisely on a line. (See graph on the previous page, bottom right.) A best-fit line can be a very useful tool for analyzing data because lines have very simple formulas describing their behavior. If, for instance, one has collected data up to time 5 and wishes to predict what the value will be at time 15, the value 15 can be inserted into the formula for the line to derive an estimation. One can also determine how good an estimate is likely to be by computing the correlation factor for the data. The correlation factor is a quantity that measures how close the set of data is to being linear; that is, how good a "fit" the best-fit line actually is. One of the most common uses of graphs is to display the information encoded in a function. A function, informally speaking, is an operation or rule that can be applied to numbers. Functions are usually graphed in the cartesian plane (that is, the x,y -plane) with the horizontal or x -axis representing the input variable and the vertical or y -axis representing the output variable. The graph of a function differs from the other types of graphs described so far in that all the points on the graph represent actual information. A concrete relationship, usually given by a mathematical formula, connects the two objects being analyzed. For example, the "squaring" function takes numbers and squares them. Thus an input of the number 1 corresponds to an output of 1; an input of 2 corresponds to an output of 4; an input of −7 corresponds to an output of 49; and so on. Therefore, the graph of this function contains the points (1, 1), (2, 4), (−7, 49), and infinitely many others. Does the point (10, 78) lie on this graph? To determine the answer, examine which characteristics all the points on the graph have in common. Any point on the graph of a function represents an input-output pair, with the x -coordinate representing input and the y -coordinate representing output. With the squaring function, each output value is the square of the corresponding input value, so on the graph of the squaring function, each y -coordinate must be the square of the corresponding x -coordinate. Because 78 is not the square of 10, the point (10, 78) does not lie on the graph of the squaring function. It is traditional to name graphs with an equation rather than with words. The equation of any graph, regardless of whether it is the graph of a function, is meant to be a perfect description of the graph—it should tell the viewer the relationship between the x - and y -coordinates of the numbers being graphed. For example, the equation of the graph of the squaring function is y = x ² because the y -coordinate of any point on the graph is the square of the x -coordinate. The line that passes through the point (0, 3) and slants upwards with slope 4 (that is, at a rate of four units up for every one unit to the right) has equation y = 4x + 3. This indicates that for every point on the graph, the y -coordinate is 3 more than 4 times the x -coordinate. An equation of a graph has many uses: it is not only a description of the graph but also a mechanism for finding points on the graph and a test for determining whether a given point lies on the graph. For example, to find out whether the point (278, 3254) lies on the line y = 4x + 3, simply insert (278, 3254), resulting in the inequality 3254 ≠ 4(278) + 3. Because t


From Yahoo Answers

Question:This is just an example to help me. Say I have 36 class votes altogether and 3 students voted for a certain thing. How would I find the degree of those 3 votes to make a circle graph? I think you can make a proportion, but i forgot how to set up the left side (of the example at bottom). I know that it is x/360 but i forgot how to do the other side. But, if you have a different way, please show it too. Ex. 3/36 = x/360

Answers:Your example is perfect. Just cross multiply: 36x = 3 360 = 1080 then divide by 36 to get x = 30

Question:I'm having a little difficulty with my work here. I have to change the percentage of a gas in the earth's atmosphere into degrees so I can put it in a pie chart. They didn't give me the formula for this, and I can't remember it for my life. One example: Gas: Nitrogen % of Atmoshpere: 78.08 I need to find the degrees of circle. If anyone can, please explain in some detail how to do this and give me an easy formula to work with.

Answers:OK, I'm not sure about a formula, but I'm sure I can get the answer. A circle is 360 degrees; so, you multiply 360 by .7808, and you get 281.088 degrees. In mathematical form: 360 x .7808 = 281.088 There's your answer. Also, you should try and learn the concept, so you're not getting your work solely from others.

Question:Also how do I convert 2/5 to degreemeasure for a circle graph?

Answers:360 degrees in a circle... How about 87% of 360?

Question:How would I be able to find the measure of the missing angle used in the construction of a circle graph? 50 , 72 , 35 , 124 , __ Thanks!

Answers:a circle equals to 360 degrees. all you need to do is add up 50, 72, 35, and 124. Adding those up would equal to 281 degrees. Its simple from there. just do 360 minus 281 and you would get 79 degrees. 79 degrees is the ANSWER. the missing angle is 79 degrees. hope this helped!

From Youtube

Trig: The Unit Circle, Degrees & Radians :www.mindbites.com This 80 minute trigonometry lesson is a study of the Unit Circle (a circle with a radius of one unit) and its x and y coordinates. This lesson will show you how to find the exact value of the six trig ratios of: - all quadrantal angles 0 or 2 Pi radians, Pi/2 radians, Pi radians, 3Pi/2 radians, - angles with a reference angle of Pi/4 radians (45 degrees) - angles with a reference angle of Pi/6 (30 degrees) - angles with a reference angle of Pi/3 (30 degrees) Thislesson contains explanations of the concepts and 12 example questions with step by step solutions plus 5 interactive review questions with solutions. Lessons that will help you with the fundamentals of this lesson include: - 300 Basic Trigonometry Part I (www.mindbites.com - 315 Basic Trigonometry Part II (www.mindbites.com - 320 Trigonometry on the Coordinate Plane (www.mindbites.com - 330 Arc Length Solving 1st Degree Trig Equations (Radians) (www.mindbites.com

Graphing Circles :www.gdawgenterprises.com This video shows how to graph circles starting with the standard form for the equation of a circle. From the standard form, algebra is used to isolate the y variable, then graphed. Special emphasis is placed on the h and k constants. The x minus a number moves the circle to the right and x plus a number moves the circle to the left. The y minus a number moves a circle up and the y plus a number moves a circle downward. The conics application in the TI 84 series calculators is used to check answers. Example problems are given as well. Background music Drifting by Andy McKee