In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection.
Plane graphs can be encoded by combinatorial maps or rotation systems.
An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a planar map. Although a plane graph has an external or unbounded face, none of the faces of a planar map has a particular status.
Planar graphs generalize to graphs drawable on a surface of a given genus. In this terminology, planar graphs have genus 0, since the plane (and the sphere) are surfaces of genus 0. See "graph embedding" for other related topics.
Homework Statement
Prove that no more than 3 edges can be added to G while keeping it planar and simple.
(The graph in the image is graph G)
Homework Equations
For a connected, planar, simple graph G, with E edges and V vertices:
E ≤ 3V - 6
The Attempt at a Solution
At first, we have 9...
Homework Statement
In an experiment, I measured the effect of mass on the acceleration of a cart, rolling down a leveled air track.
I then graphed acceleration vs. mass (m'), where:
m' = m / (M+m); M = mass of the cart, m = hanging mass, which was increased with subsequent trials.
Homework...
1. I'm given time (s), extension (mm), tensile stress (mPa), and tensile strain (%) in a table. I'm supposed to graph it on excel, but my graphs don't look at all like a stress-strain graph.
2. I think it's probably because I'm using tensile stress or strain? Or is it an excel error I'm doing...
Homework Statement
See attached image.
Match the type of system or situation to the appropriate energy level diagram.
1) hadronic (such as +)
2) idealized quantized spring-mass oscillator
3) nuclear (such as the nucleus of a carbon atom)
4) vibrational states of a diatomic molecule such as O2...
I would like to plot a number of 3D graphs from different data files. For example I am using
fid = fopen('SS 1.dat','r');
to read the first file and then plot a graph. How to set the program to change the name to 'SS 2.dat' automatically? Also for the tenth file the name becomes 'SS 10.dat'...
How do you find a normal vector of a function at a point, such as f(x,y)= ax^y+yx^y^x+b at (X_o,Y_o)
where a and b are just arbitrary constants, and the function is an arbitrary function. So I guess, what is the general steps you take to find the normal? I thought it had to do with the...
Homework Statement
A 1D wave function ψ(x,t) satisfies these initial conditions:
ψ(x,0) = 0 for all x
∂ψ/∂t (x,0) is v for -a≤x≤a
0 otherwise
Plot ψ(x,t) as a function of x at time t=a/v.
Homework EquationsThe Attempt at a Solution
I know the 1D wave equation is given by...
Homework Statement
4. On which graph are the characteristic X-rays being portrayed correctly.
Homework Equations
The Attempt at a Solution
ok I know that characteristic X-rays happen because the electrons in outer shells move to the lowest possible energy shell and that the change in...
Homework Statement
What would the slope of a position vs time squared graph represent?Homework Equations
X=my^2+bThe Attempt at a Solution
I thought the slope of this graph would be velocity due to the fact that the normal position vs time graph slopes are velocity. However, I was told that the...
So this question is from a physics lab. The apparatus for the experiment was made upon a device that measures angular speed over time. On top of this is a rail to which different items can be attached. Directly in the center of the rail is a device which has a pulley with a string to attached...
Homework Statement [/B]
Don't know if this goes here or in the advanced bit, thought I'd try here first!
I know the general solution of a 1D wave equation is given by d'Alembert's formula
##u(x,t) = 0.5[u(x+vt,0) + u(x-vt,0)] + \frac{1}{2v} \int_{x-vt}^{x+vt} \frac{\partial u}{\partial...
Homework Statement
Parts which I am not sure about are in bold.
(a) A car of mass 1000 kg is initially at rest. It moves along a straight road for 20 s and then comes to rest again. The speed-time graph for the movement is:
(i) What is the total distance travelled?
(ii) What resultant force...
Homework Statement
Hi,
I am stuck with the following question:
The graph (attached) shows how the Moon's gravitational pull F on a lunar lander varies with the distance 'h' from the Moon's surface. Estimate the work done by the pull of the Moon on the lunar lander as it approaches the lunar...
Homework Statement
the ans given is 95Nm , but i gt 80.9Nm , which part i did wrongly ?
Homework EquationsThe Attempt at a Solution
20(4) -39(5/surd 26(3)) +39(1/surd26)(1) -60(3/5)(1)+60(4/5)(3)
=80.9[/B]
The initial position of the block is the origin; i.e., x = 0 at t = 0 . Consider down the track to be the positive x-direction. A block with an initial velocity v0 slides up and back down a frictionless incline. Which graph best represents a description the position of the block versus time...
Homework Statement
A ball is thrown vertically upwards and caught by the thrower on its return. Sketch a graph of velocity (taking the upward direction as positive) against time for he whole of its motion, neglecting air resistance. How, from such a graph, would you obtain an estimate of the...
Homework Statement
A student investigates the motion of a trolley as it rolls down a ramp ... a data logger records the time the beam was broken for and uses this to calculate the speed of the trolley.
Experiment results are:
Height of ramp (cm) / Momentum of trolley at the bottom
15...
I am searching for a site that enables me to plot a function of multiple variables in 3D form. Say f(x1,x2,x3,x4) and I wish to plot this function as a 3D graph for x1 and x2 where x3 and x4 are kept constant.
Homework Statement
The figure below represents part of the performance data of a car owned by a proud physics student. (The horizontal axis is marked in increments of 2 seconds and the vertical axis is marked in increments of 10 m/s.)
(a) Calculate the total distance traveled by computing the...
I have a graph of force vs. acceleration of a kart on a a ramp that i manually pushed back and forth with my hand.
The mass of the the kart is .6485kg and when i did the linear fit it said that slope of the graph was .5586N/m/s^2 and the intercept .2097 N
My question is would i see the mass of...
Homework Statement
1. Consider the Cartesian Product A X B, where A, B are finite nonempty sets, each with carnality greater than 1. There are two functions with domain A X B, called projections with mapping rules p1(a,b) = a and p2(a,b) = b. What is the target space of p1? p2? Are either of...
I'm not sure if this warrants a full post, but I am doing my homework and I came across notation I'm not familiar with. Skimming the chapter it's not in there either.
It says "Draw W6" but W6 has a bar over it, like complex conjugate. What does this mean? I know what W6 looks like.
Dear Physics Forum friends,
I am a college junior who is currently conducting the undergraduate research in the theoretical computer science. I wrote this post to seek you recommendation on the books that separately treat the graph theory and combinatotics, both in theory and applications. I...
Homework Statement
Ok.Then there was a question
If r is distance measured from center of a charged shell and R is it's radius then the graph which may correctly represent variations of electric field is
Homework Equations
E vs R graph
The Attempt at a Solution
The answer is given to be A...
Hi,
Please note: this is not a homework question! It is a real world problem I am trying to solve.
I have some values in mA and tonnes which I need to extrapolate but they are not linear. I know it the mA curve drops off the higher the tonne values go.
I have plotted the values in excel...
I am a new university physics student doing some homework, but i cannot find the method for this problem.
How do I find the magnitude of a force from an acceleration vs. mass graph?
I know mass is the inverse of the slope of an acceleration vs. force graph, but I'm not sure how to do this the...
Homework Statement
Determine from the left graph the speed at t1.
Answer:
Draw the tangent (right graph) at t1 and determine the slope. v = 1.4/0.30 = 4.7 m/s
2. The attempt at a solution
The slope is (Y2-Y1)/(X2-X1). t1 coordinates are (0.24-0.25, 0). In order to get the answer I should...
Hi!
I have just performed some experiments with my phones accelerometer, and plotted the acceleration as a function of time from the raw data. The experiments I did was:
jumping down from a chair
jumping from the floor and up
let the device fall toward a bed
When I analyze the graphs, I think...
So we started looking at Zener Diodes in our electronics course.
What I am having trouble with is the interpretation of the graph.
For the example our diode has a Zener Voltage is 6.2V. I understand what happens in the positive half but when the direction changes is where I get confused.
Why...
I greatly appreciate this chance to submit a query.
I have the following integral: $$\int_{1}^t 2sin(t-\tau)e^{-2(t-1)} d\tau$$
and it has been suggested to me that if I sketch the two constituent functions and multiply them, I can read the answer off the paper. So here are my sketches: go...
Homework Statement
Ok, so I did this..
V=IR
I=(1/R)V, gradient is (1/R) -> y=mx+c
So in order to have the smallest resistance, we choose the point on the graph that has the largest gradient, which I chose B, because the tangent on the graph on point B is the largest, but the correct answer is...
Homework Statement
[/B]
Take the slope of the curve in the figure to determine the velocity at t = 10.0 s, in meters per second.
Homework Equations
(y2-y1)/(x2-x1)[/B]The Attempt at a Solution
I don't know how to find it at 10 because there isn't a point plotted at 10. I did the two...
I feel like this should be a super easy problem, but I'm not understanding something about it.
The graph of y = f(x) passes through the points (0, 1), (1, 2), and (2, 3). Find the corresponding points on he graph y = f(x + 2) - 1. I graphed the (x + 2) - 1, but what does it mean by...
Greetings, i found an interesting exercise from my perspective, it's not about HW, i just want to see different approaches than the Greek math forum i posted yesterday, so we have:
If $$ f $$ is a function, then a chord is a straight portion whose edges belong to $$ C_f $$
f is a continuous...
Homework Statement
A motorcycle cop, parked at the side of a highway reading a magazine, is passed by a woman in a red Ferrari 308 GTS doing 176. km/h. After a few attempts to get his cycle started, the officer roars off 2.00 s later and catches the Ferrari at the state line 2.00 km away some...
What software is used to create a text editor, math graph, and math equation such as below image?
It seems the math graph is a vector illustration (not pixel), the math equation is not Latex, and they are integrated with the text editor.
What software to create such as book?
Homework Statement
Hello.
The resistance of a wire increases when the temperature increases.
How would this cause the shape of the graph to change? Explain why. The graph shown is a straight line graph with the line traveling at 45 degrees path. The x-axis is VOLTS. The y-axis is CURRENT...
Homework Statement
A boat is sailing starting from x = 200m.
When interpreting x(position) vs. t(time) graph for this boat with math models x = -20t + 200(t<=10),
Is it right to say "The boat is approaching to the origin" ? (When t <= 10)
Homework Equations
x = -20t + 200(t<=10)
The...
How many trees are in a graph if the number of nodes and the number of branches (or meshes) is given? I refer to the trees which has to "walk" through the nodes along the given branches (so no tree between 2 nodes without a connecting branch between them)
MODERATOR'S NOTE: THIS WAS MOVED FROM GENERAL PHYSICS TO HOMEWORK FORUM, SO THERE IS NO TEMPLATE
Suppose a book is given a force F across a table with with friction. It will have an initial velocity v until it comes to rest due to friction. Does anyone know what will be the graph of kinetic...
I've been really into electrons and their orbitals for a few months now, but I've never understood how they come up with all of these 3d models from these complicated equations. I would love it if someone could explain in detail the equations and the values of the variables and constants within...
Homework Statement
Calculate the average velocity after 11 s.
Homework Equations
v=d/t
The Attempt at a Solution
I need to find the total displacement to find average velocity. I assumed the displacement was -4 as the graph goes up then back down to 0 then -4 so total is just -4. Is that a...