circuits, paths, and graph structures packet

AQR UNIT 7

NETWORKS AND GRAPHS:

Circuits, Paths, and Graph Structures

Packet #___

BY:_____________________

IntroductiontoNetworksandGraphs:Trydrawingapathforapersontowalkthrougheachdoorexactlyoncewithoutgoingbackthroughanydoormorethanonetime.Try1: Try2:

__________________________________________________________________________________________________________________Creatingefficientroutesforthedeliveryofgoodsandservices(suchasmaildelivery,garbagecollection,policepatrols,newspaperdeliveries,andlate-nightpizzadeliveries)alongthestreetsofacity,town,orneighborhoodhaslongbeenaproblemforcityplannersandbusinessesalike.ThesetypesofmanagementscienceproblemsareknowninmathematicsasEulercircuitproblems.Eulercircuitproblemscanallbetackledbymeansofasingleunifyingmathematicalconcept–theconceptofagraph.Themostcommonwaytodescribeagraphisbymeansofapicture.Thebasicelementsofsuchapictureare:asetof“dots”calledtheverticesofthegraphandacollectionof“lines”calledtheedgesofthegraph.InanEulercircuitproblem,bydefinitioneverysingleoneofthestreets(orbridges,orlanes,orhighways)withinadefinedarea(beitatown,anareaoftown,orasubdivision)mustbecoveredbytherouteexactlyonce,endingatthelocationusedasthestart.__________________________________________________________________________________________________________________Hereisaselectionofgraphs.Aretheythesameordifferentthaneachother?Whatmakesagraphdistinctfromanother?

Partsofagraph:Labeleachgraphwithitsuniquefeature.

Student Worksheets Created by Matthew M. Winking at Phoenix High School SECTION 7-3 p.90

Sec 7.3 – Euler Circuits & Paths Networks & Graphs Name:

Some brainteaser problems involving networks like the following simply cannot be done. The famous Konigsberg bridge problem is one of them. It is not possible to find a path that will enable you to cross each bridge exactly once (meaning any path that a person walks over every single bridge will require that person to walk across at least one bridge more than once). Another fairly common impossible problem is drawing a path for a person to walk through each door exactly once without going back through any door more than once.

(Can you prove Leonhard Euler wrong? Can you find a path to walk that only takes you over each bridge JUST ONCE?

(Can you prove Leonhard Euler wrong? Can you find a path to walk that only takes you each door JUST ONCE?)

(Can you prove Leonhard Euler wrong? Can you find a path to walk that only takes you over each bridge JUST ONCE?)

If the following graphs can be created without picking up your pencil and without ever retracing any edge, the graph is said to be traversable of these some are referred to as Euler Circuits or Euler Paths. Can you determine which are traversable? Circle the ones that are: Example that is traversable. v

Koenigsberg Bridge Problem The Impossible

Doorway Problem

Alternately, you can label each vertex and then list the vertices in order that would complete the

graph.

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Sec 7.3 – Euler Circuits & Paths Networks & Graphs Name:

Some brainteaser problems involving networks like the following simply cannot be done. The famous Konigsberg bridge problem is one of them. It is not possible to find a path that will enable you to cross each bridge exactly once (meaning any path that a person walks over every single bridge will require that person to walk across at least one bridge more than once). Another fairly common impossible problem is drawing a path for a person to walk through each door exactly once without going back through any door more than once.

(Can you prove Leonhard Euler wrong? Can you find a path to walk that only takes you over each bridge JUST ONCE?

(Can you prove Leonhard Euler wrong? Can you find a path to walk that only takes you each door JUST ONCE?)

(Can you prove Leonhard Euler wrong? Can you find a path to walk that only takes you over each bridge JUST ONCE?)

If the following graphs can be created without picking up your pencil and without ever retracing any edge, the graph is said to be traversable of these some are referred to as Euler Circuits or Euler Paths. Can you determine which are traversable? Circle the ones that are: Example that is traversable. v

Koenigsberg Bridge Problem The Impossible

Doorway Problem

Alternately, you can label each vertex and then list the vertices in order that would complete the

graph.

A

B

C

D

H

G

F

E

I

K L

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Advanced Mathematical Decision Making (2010) Unit VII, Section A Planner

Charles A. Dana Center at The University of Texas at Austin

VII-6

functions. As with other mathematical models, graphs are used to translate

mathematical procedures (arithmetic and pattern recognition mostly) in order

to shed light on real-world or physical problems. This section also gives students the

opportunity to decide on their own how to use a graph to model the given situation.

Encourage students to pay careful attention to definitions and form their own

conjectures and mathematical questions about graphs.

As conjectures arise, students should think about testing them with a variety of

examples. Coming up with counterexamples to disprove a conjecture should be seen

as healthy competition and not as a way to destroy another student’s idea.

Things to Watch for

As mentioned previously, the use of a graph in this unit is not to be confused with the

use of a graph in the context of graphing functions. This meaning of the word graph

is entirely different from its meaning in “the graph of a function.” Care needs to be

taken when this terminology is used for the first time. This lesson also provides

definitions for vertex, edge, path, circuit, Euler circuit, and Hamiltonian circuit.

There are many ways to represent the same graph. For example, the following are all

considered the same graph because they each consist of two vertices connected by a

single edge:

Some graphs contain loops, edges that begin and end at the same vertex. If students

bring up this idea, mention that mathematicians certainly work with such graphs, but

this unit does not. The following is a graph with a loop:

Sometimes multiple edges between the same two vertices are allowed. When making

conjectures, it must be clear if such things as multiple edges and loops are being

considered.



VII-6










Things to Watch for








single edge:






considered.



VII-6










Things to Watch for








single edge:






considered.

Modelinggraphs:Inthespacebelowdrawagraphofthefollowingscenario:Youand3friendsallliveindifferenthousesinthesameneighborhood.Eachhouseisconnectedbyroadstoeachoftheotherhouses.

Countinggraphs:Agraphcanbeidentifiedbyitsedgesandverticesandtherefore,bythedegreeofeachvertexpoint.Drawagraphwithonlyevendegreeverticesandagraphwith2oddverticesandtheresteven.Thencountthedegreesofeachvertexinthepictureontheright.

Walks,Paths,Circuits:

a)Tracethepictureattherightwithoutpickingupyourpencil.b)Tracethepictureattherightwithoutpickingupyourpencilorretracinganysteps(edges).c)Tracethepictureattherightwithoutpickingupyourpencilorretracinganysteps(edges),endingwhereyoustarted.

EulerCircuits:Attempttotracethefollowingshapeswithoutliftingyourpencilorretracinganysteps(edges)andendingwhereyoustarted.

� Sometimes edges “cross” each other at incidental crossing points that are not themselves vertices of the graph. !

Such is the case with the crossing point created by edges AD and BE .

Example Connect the Dots 18!

2 INTRODUCTION TO GRAPH THEORY WORKSHEET

So, has your group come to a conclusion? You’ve probably had a hard time finding a routethrough across all seven bridges. If you think that such a path cannot exist, we need tounderstand why it cannot. That is, we need to prove that no solution exists.

2. Graph Theory Definitions

Define the following graph theory terms using complete sentences.

Vertex (Vertices):

Edges:

Graph:

Path:

Circuit:

Eulerian Path and Eulerian Circuit:

Now that we’ve learned some graph theory we can re-envision the shapes from the beginningof class as graphs. Note, we could have added more (or less) vertices than we have, but thefollowing graphs should su�ce.

Filloutthechartbasedonthepreviousshapes.Forthischart,traceablemeansgraphsthathaveanEulercircuit

KönigsbergBridge:ThefollowingfigureshowstheriversandbridgesofKönigsberg.Residentsofthecityoccupiedthemselvesbytryingtofindawalkingpaththroughthecitythatbeganandendedatthesameplaceandcrossedeverybridgeexactlyonce.IfyouwerearesidentofKönigsberg,wherewouldyoustartyourwalkandwhatpathwouldyouchoose?Note,therivercutsthecityinhalf,soonecannottravel“outside”ofthepicturetogetfromthebottomhalftothetophalf.

RepresenttheKönigsbergBridgeasagraph:

Alternate Versions of Task

More Accessible Version:

Try tracing the figures below without lifting your pencil from the paper, without retracing anylines or crossing any lines. Some figures you can trace and some figures you cannot trace.Euler, an early mathematician, studied figures like these and came up with a rule so that youwould know which figures could be traced and which could not. He found it helpful to categorizevertices as odd or even depending on the number of line segments coming from each vertex.Investigate figures to see if you can come up with Euler's rule for classifying figures astraceable and those that are not traceable. Record your findings in the chart below to organizeyour information in a way that helps to draw conclusions about Euler’s rule.

Diagram # Traceable? # of Odd Vertices # of Even Vertices Total Vertices1.2.3.4.5.6.7.8.9.

10.

Your conclusion:

More Challenging Version:

Have students go to the following web site to solve Euler’s famous Seven Bridges of Konigsbergproblem: http:mathforum.org/isaac/problems/bridges1.html

Context

The class was measuring and classifying angles and naming polygons. We were also workingon looking for evidence in student work to begin self-assessing problem solving. I had thestudents thinking of themselves as investigators looking for evidence in their work, as well asother student work. This problem seemed to go along with our investigative work from adifferent angle. Instead of looking for evidence, they needed to produce the evidence.

What This Task Accomplishes

This task allows me to determine which students can experiment, record results and look forpatterns when what they are looking for is obscure. A student will need a way to organize theresults of their experimenting carefully so they can begin to see patterns.

4 of 12Euler's DilemmaCopyright ©, 2005. Exemplars. All rights reserved.

INTRODUCTION TO GRAPH THEORY WORKSHEET

1. The Seven Bridges of Konigsberg

You and your group member are all students of mathematics in Konigsberg, and the yearis 1735. Your city has a river that runs through it, and like many of the inhabitants ofyour city, you all try to find a route through yhe city (pictured below) that crosses eachof the city’s bridges exactly once. Unlike most of the cities inhabitants, however, you areall mathematician, and are trying to understand the question mathematically. Once more,you’ve all heard that the famous Swiss mathematician Leonhard Euler will be arriving intown to speak at your school in the next few days. Before he arrives you’ve all agreed tosolve the ”Seven Bridges of Konigsberg question,” to find a route across all seven bridges,or to explain why such a route cannot exist.

Directions: First, to get familiar with the problem, try to find a path throughthe city of Konigsberg that crosses each bridge only once. Do you think it canbe done? Note, the river cuts the city in half, so one cannot travel “outside” ofthe picture to get from the bottom half to the top half.

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Problems:Completethefollowingproblems.1)WhataboutwhenyouvisittheEasternandWesternwildflowergardensthathavefabuloussculpturesinadditiontobeautifulflowersalongthewalkways.Youwanttoseeeachdisplaywithoutbacktracking(seeingsomethingyouhavealreadyseen).Wherewouldyoustartyourwalkandwhatpathwouldyouchoose?

2)YourfriendChetcallsyouonhiscellphoneandtellsyouthathehasdiscoveredalargerockembeddedwithgems!Heissomewhereinyourfavoritehikingarea,whichhasmanyinterconnectedpaths,asshownbelow.Chetdoesnotknowexactlywhereheis,butheneedsyourhelptocarrytherock.Tofindhim,youdecideitwouldbemostefficienttojogalongallthepathsinsuchawaythatnopathiscoveredtwice.Findthisefficientrouteonthemapbeloworexplainwhynosuchrouteexists.

Teacher Version

Networks and Graphs: Circuits, Paths, and Graph Structures

VII.A Student Activity Sheet 1: Euler Circuits and Paths


Advanced Mathematical Decision Making (2010)

Activity Sheet 1, 8 pages

VII-22

2. What about when you visit the Eastern and Western wildflower gardens that have

fabulous sculptures in addition to beautiful flowers along the walkways. You want

to see each display without backtracking (seeing something you have already seen).

Where would you start your walk and what path would you choose?

Western garden

Eastern garden

It is not possible to tour the Western garden without backtracking because of the three exhibits on the periphery. In the Eastern garden, a path exists, but the entry and exits points are different.

Teacher Version

Networks and Graphs: Circuits, Paths, and Graph Structures

VII.A Student Activity Sheet 1: Euler Circuits and Paths




VII-26

8. What does your conjecture tell you about the Königsberg Bridge problem and the garden scenario?

The graph representing Königsberg has vertices with an odd degree (in fact, all of the vertices have an odd degree); therefore, a Euler circuit does not exist. This means that nobody can find a path starting and stopping at the same place and traversing all seven bridges exactly once. In the Western garden, four vertices have an odd degree, including the three vertices on the periphery; therefore, no path exists. The Eastern garden has one vertex with an odd degree; therefore, no Euler circuit exists. A Euler path exists in the Eastern garden, but it does not begin and end at the same vertex.

9. Your friend Chet calls you on his cell phone and tells you that he has discovered a large rock embedded with gems! He is somewhere in your favorite hiking area, which has many interconnected paths, as shown below. Chet does not know exactly where he is, but he needs your help to carry the rock. To find him, you decide it would be most efficient to jog along all the paths in such a way that no path is covered twice. Find this efficient route on the map below or explain why no such route exists.

If you model this map as a graph with the paths representing edges and the intersections (or corners) being the vertices, every vertex has an even degree; therefore, there must be a Euler circuit. Starting in the upper left and following the edges in increasing order is one example of a Euler circuit.

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3)Youhavebeenhiredtopainttheyellowmedianstripeontheroadsofasmalltown.Sinceyouarebeingpaidbythejobandnotbythehour,youwanttofindapaththroughthetownthattraverseseachroadonlyonce.Inthemapofthetown’sroadsbelow,findsuchapathorexplainwhynosuchpathexists.

4)Isitpossibletodrawapathforapersontowalkthrougheachdoorexactlyoncewithoutgoingbackthroughanydoormorethanonetime?Ifso,showthepaththendetermineifitispossibletodosoandendbackattheplaceyoustarted.

Teacher Version

Networks and Graphs: Circuits, Paths, and Graph Structures VII.A Student Activity Sheet 1: Euler Circuits and Paths




VII-27

10. You have been hired to paint the yellow median stripe on the roads of a small town.

Since you are being paid by the job and not by the hour, you want to find a path

through the town that traverses each road only once. In the map of the town’s roads

below, find such a path or explain why no such path exists.

Again, if you model the street system as a graph, then this graph has several vertices of an odd degree. For example, there are two vertices of Degree 3 along the top of the graph. This means that a Euler circuit does not exist; you cannot travel through the entire town by traversing each street exactly once.

11. REFLECTION: For what situation(s) is it satisfactory to have only a path exist and not a

circuit?

Answers may vary. Sample student response:

In any situation that can be traversed (all edges and all vertices only once) and entry and exit points do not necessarily coincide, a graph that is a path exists. If this is satisfactory, then a circuit is not necessary.


1. a. Label the degree of each vertex b. Put a CIRCLE around the following graphs that have an EULER CIRCUIT and list a possible circuit.

Briefly explain why an Euler Circuit must have all even degree vertices.

c. Put a SQUARE around the following graphs that have an EULER PATH and list a possible path. Briefly explain why an Euler P must have exactly 2 odd vertices and the rest even.

2. Create a Graph of the following map and explain whether it is impossible or possible to pass through

each door exactly once.

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ROOM A

ROOM B ROOM C ROOM D

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OUTSIDE

Follow-upQuestions:1)

2)FormaconjectureabouthowyoumightquicklydecidewhetheragraphhasanEulercircuit,andexplainwhyyourconjectureseemsreasonable.WhatdoesyourconjecturetellyouabouttheKönigsbergBridgeproblemandthegardenscenario? 3)REFLECTION:Forwhatsituation(s)isitsatisfactorytohaveonlyapathexistandnotacircuit?

2 INTRODUCTION TO GRAPH THEORY WORKSHEET

So, has your group come to a conclusion? You’ve probably had a hard time finding a routethrough across all seven bridges. If you think that such a path cannot exist, we need tounderstand why it cannot. That is, we need to prove that no solution exists.

2. Graph Theory Definitions

Define the following graph theory terms using complete sentences.

Vertex (Vertices):

Edges:

Graph:

Path:

Circuit:

Eulerian Path and Eulerian Circuit:

Now that we’ve learned some graph theory we can re-envision the shapes from the beginningof class as graphs. Note, we could have added more (or less) vertices than we have, but thefollowing graphs should su�ce.

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circuits, paths, and graph structures packet

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