network math

13The river Pregel runs through the university town of Königsberg (now Kaliningrad in Russia). In the middle of the river are two islands connected to each other and to the rest of the city by 7 bridges. Many years ago, a tradition developed among the university students where they challenged one another to make a round trip over all 7 bridges, walking over each bridge once, and only once, before returning to the starting point. Can it be done? Try different trips and record your results.

In this chapter you will work with a number of networks and look at a rule to determine whether a network can or cannot be ‘travelled’.

Networks

MQ 8 Ch 13 Page 531 Friday, December 8, 2000 8:02 AM

532 M a t h s Q u e s t 8 f o r V i c t o r i a

What is a network?People, places and events are all connected. Networks help us to understand connectionsand in this chapter we will look at the properties and applications of networks. Euler’sformula that has been previously discussed, will be investigated in relation to networks.

The last three sections ofthe chapter are devoted tostudies of paths, circuits andtrees. In those sections wewill investigate methods offinding Euler and Hamilto-nian paths and circuits, andminimal spanning trees forvarious networks. We willalso learn how these tech-niques can be applied in prac-tical situations and forproblem solving.

Note: Some materialcovered in this chapter canbe treated as an extension, oradvanced material, which isbeyond the scope of a typicalYear 8 class.

What does the Internet have in common with the roads in a suburb? They both canbe considered as networks. What is a network?

A network is a collection of objects, connected to each other in some specific way. The objects are called vertices (or nodes) and the connections are called edges.

The Internet is a collection of computers, connected to each other, while the roads ina suburb connect all the houses, parks and other features of a suburb.

Even the structure of your family (your family ‘tree’) can be considered as a network.

A simple networkConsider the case of Adam and Betty who are married. We can rep-resent this family as a network, as shown in the figure on the right.The circles (vertices) represent the people, while the line (the edge)indicates that there is a ‘connection’ by marriage.

Later on they decide to have children, and have a boy, Cecil, and agirl, Daphne. The network now looks like the figure on the right. Theadditional lines joining A to C and D, and B to C and D indicate a‘connection’ between parents and children.

Later on, Daphne decides to get married to a man namedEric; so now the network would look like the figure on the right.

The line joining D to E indicates their marriage.

A B

A B

C D

A B

C D E


C h a p t e r 1 3 N e t w o r k s 533We could continue this process indefinitely, creating a family tree structure.Although it does not look like the ‘traditional’ family tree diagram, (as shown

below), it is equivalent to it.

In this case the objects in our network are the people and the lines indicate some‘relationship’ between them.

A B

C D E

Draw the network which represents the family tree showing Alice, her two parents Bill and Carla, and her grandparents, David and Eunice (paternal), and Frank and Gemma (maternal).

THINK WRITE/DRAW

List the objects (people) in the network. A for Alice, B for Bill, C for Carla, D for David, E for Eunice, F for Frank, G for Gemma.

Draw them using the vertical direction to indicate the ‘generations’.

Join the various people with lines, representing parents and marriage.

D is married to E.

F is married to G.

D and E are the parents of B.

F and G are the parents of C.

B is married to C.

B and C are the parents of A.

1

2 D E F

B C

A

G

D E F

B C

A

G

1WORKEDExample



Representing a networkAs we have seen, a network is completely determined by the vertices and the edgeswhich connect the vertices, and can be represented by the diagram. It can also bedescribed using what is known as the labelling method.

A network can be represented by a list of pairs of vertices. The vertices in every pair are connected by an edge. This is called the labelling method.

Represent the network shown at right, using the labelling method.

THINK WRITE

Start with any vertex, say, A. Vertex A is connected to B and to C. List each connection as a pair.

Vertex A: (A, B), (A, C)

Vertex B is connected to A, C, D and E. List each connection as a pair, but do not repeat any connection that has already been listed. Since the connection between A and B was listed as (A, B) in step 1, do not list (B, A).

Vertex B: (B, C), (B, D), (B, E)

Continue for each remaining vertex:Vertex C is connected to A, B, F, G but do not repeat (C, A), (C, B).Vertex D is connected to B and E but do not repeat (D, B).Vertex E is connected to B and D, but these were already listed as (B, E) and (D, E). So do not list any pairs for E.Vertex F is connected to C and G but do not repeat (F, C).Finally, vertex G is connected to F and C, but these were listed as (C, G) and (F, G). So do not list any pairs for G.

Vertex C: (C, F), (C, G)

Vertex D: (D, E)

Vertex E: has no other connection.

Vertex F: (F, G)

Vertex G: has no other connection.

Put all the pairs into a single list. {(A, B), (A, C), (B, C), (B, D), (B, E), (C, F), (C, G), (D, E), (F, G)}

1

2

3

4

2WORKEDExampleD E F

B C

A

G


C h a p t e r 1 3 N e t w o r k s 535There should be exactly one pair for each edge in the network. For worked example 2there are 9 edges and 9 pairs. How many edges were there in worked example 1? It isoften more practical to go from the labelled network to a network drawing, as shown inthe next worked example.

Of course, it is possible to get many different ‘shapes’ for your drawing, but theconnections (edges) must still be correct.

Draw the network represented by the following set of labels.{(A, B), (A, C), (B, C), (B, D), (C, D), (C, E), (D, E)}

THINK DRAW

Draw vertex A. By studying the set, observe that A is contained in the labels (A, B) and (A, C), which tells us it is connected to B and C. So draw vertices B and C and connect them to A.Cross off the labels that have been used. (Once we have used the label, there is no need to go back to it.) Vertex B is included in (B, C), so add the connections from B to C. It is also included in (B, D). So add the vertex D, and the connections from B to D. Cross off the labels (B, C) and (B, D).Vertex C is included in (C, D), so add the connection from C to D. It is also included in (C, E). So add vertex E, and the connection from C to E. Cross off the labels (C, D) and (C, E).

Vertex D is included in (D, E) and hence is connected to E. So add the appropriate connection.

1 A B

C

2 A B

C

D

3 A B

C

E

D

4 A B

C

E

D

3WORKEDExample

remember1. A network is a collection of objects connected to each other in some specific

way. The objects are called vertices (or nodes) and the connections are called edges.

2. A network can be represented by a list of pairs of connected vertices. This is called the labelling method.

remember

MQ 8 Ch 13 35 Friday, December 8, 2000 8:02 AM


What is a network?

1 Draw the network, which represents the following family tree.Henry and Ida marry and have a single child, Jane.Jane marries Kenneth and they have a single child, Louise.Louise marries Mark and they have two children, Neil and Otis.

2 Four towns, Joplin, Amarillo, Flagstaff and Bairstow are connected to each other asfollows:

Joplin to Amarillo; Joplin to Flagstaff; Amarillo to Bairstow; Amarillo to Flagstaff;Flagstaff to Bairstow.

Draw the network represented by these connections.

3 A fifth town, Kingman, is added to the network of question 2, with roads connectingit to Amarillo, Flagstaff and Bairstow. Modify your drawing, adding the new vertexand its connections.

4 Represent the network shown, using the labelling method.

5 Represent the network shown in the following diagrams, using the labelling method.a b c

6 Draw the network represented by the following set of labels.{(A, B), (A, C), (B, D), (C, D), (C, E), (D, E)}

7 Draw the network represented by the following.{(A, B), (A, C), (A, D), (B, E), (C, E), (C, F), (C, G), (D, G)}.

8 Draw the network represented by the following list.{(A, B), (A, C), (B, C), (B, E), (C, D), (C, E), (C, F), (D, E), (E, G), (F, G)}.

9A network consists of vertices and:

10Which of the following is not a network?

A letters B edges C nodes D lists E none of the above

A The telephone system of BrisbaneB The railway system of SydneyC The family tree of the Queen

of EnglandD The history books in a libraryE The trails in a national park

leading to tourist attractions

13AWWORKEDORKEDEExamplexample

1

A B

C D

WWORKEDORKEDEExamplexample

2

E C A

D B

A B C

DEF

R V

UTQ

SP


3

mmultiple choiceultiple choice



C h a p t e r 1 3 N e t w o r k s 537

Basic properties of networksUp until now we have seen networks only as a collection of vertices (or nodes) and theedges connecting them. What are some of the mathematical properties of such networks?In this section we will build up a set of ‘rules’ and properties leading to a very specialformula for networks: Euler’s rule.

The degree of a vertexThe degree of a vertex is the total number of edges to which the vertex is connected.

The degree of each vertex can be obtained by simple counting.

There is another interesting property of the degrees in a network.

The sum of all the degrees in a network is exactly twice the number of edges.

So, for the last worked example there are 7 edges, which is half of 14, the total of allthe degrees in that network.

Planar networksA special kind of network is called a planar network, or planar graph.

A planar network is a network where the edges do not cross each other.

Most networks can be ‘fixed’ so that there are no crossing edges. An example of such a network is shown in the figure on the right.

State the degree of each vertex in the network shown.

THINK WRITE

Starting with any vertex, count its number of edges (connections). The number of edges is equal to the degree of that vertex.

Vertex A is connected to B and C, so the degree = 2.

Repeat step 1 for all remaining vertices. Vertex B is connected to A, C and D, so the degree = 3.Vertex C is connected to A, B, D and E, so the degree = 4.Vertex D is connected to B, C and E, so the degree = 3.Vertex E is connected to C and D, so the degree = 2.

1

2

4WORKEDExampleA B

C

E

D

A B

C D



If we realise that the edges do not have to be straight lines but can be curved, we canredraw the network so that it becomes a planar graph. One possible way this can bedone is shown in the figure below. Can you think of any other way?

For the rest of this section we will work with planar graphs.

Edges, faces and verticesIf you look carefully at any planar graph you will seethat the lines (edges) divide the surrounding spaceinto regions, also known as faces.Consider the network on the right. There are threeregions (labelled I, II and III). The vertices (A, B, Cand D) are not part of any region. Note: The space‘outside’ the entire network is counted as a region.(In this case it is the region labelled III.)

A B

C D

A B

C D

AIII

I

II

B

C D

Find the number of vertices, edges and regions in the network shown.

THINK WRITE/DRAW

Label the vertices and count them.

Number of vertices = 7Count the edges. The easiest way to do this is to cross off every edge that has been counted with a small mark. (This will guarantee that no edge is missed and no edge is counted twice.)

Number of edges = 10

Count the regions by ‘colouring them in’. Do not forget the region outside the network.

Number of regions = 5

1C

D

G

A

B

E

F

2

3C

D

G

A

B

E

F

5WORKEDExample



Euler’s formulaThe famous mathematician Leonhard Euler (pronounced ‘oi-ler’ or ‘uh-ler’) discovered(among other things) the relationship between the number of faces, edges and verticesfor all planar networks. This relationship is given by the following formula.

For any planar network,V = E − F + 2

where E is the number of edges, F is the number of faces (regions) and V is the number of vertices of the network.

Confirm Euler’s formula for the network shown at right.

THINK WRITE/DRAW

Count the number of vertices by labelling them.

V = 6Count the number of edges, crossing out each edge that has been counted as you go.

E = 10

Count the number of faces (regions) by colouring them in.

F = 6Write Euler’s formula. V = E − F + 2Substitute the values of the pronumerals into the right-hand side (RHS) of the formula.

RHS = 10 − 6 + 2

Simplify the right-hand side. = 4 + 2= 6

Compare the RHS with the LHS and draw your conclusion.

= LHSTherefore, Euler’s formula is confirmed.

1 D

E F

A

B

C

2

3

D

E F

A

B

C

45

6

7

6WORKEDExample



History of mathematicsLEONHARD EULER (15 Apri l 1707–18 September 1783)

During his life . . .Captain Cook lands in Australia.The sextant is invented by sea captain John Campbell.James Watt develops a more efficient steam engine.The first real restaurant opens in Paris.

Leonhard Euler was born in Switzerland and was the son of a Lutheran minister. His father wanted him to study theology but Euler was far more interested in mathematics. He attended the University of Basel from the age of 13 and received his masters degree at the age of 17. Three years later he entered the Paris Academy Prize for the first time. His entry, involving difficult calculations for the masts of ships in the Swiss Navy, received an honourable mention for its excellent mathematical methods. However, the judges were surprised that he did not seem to understand that Switzerland, being surrounded by land, did not actually have a navy! He went on to win the prize on 12 other occasions during his life.

Euler moved to Russia in 1727 where he became professor of physics and of

mathematics at the Academy of Sciences in St Petersburg. When he was 31, Euler lost the sight of one eye, possibly due to overwork. He left Russia in 1741 to become professor of mathematics at the Berlin Academy of Sciences. In 1766 he returned to St Petersburg. Shortly afterwards he lost the sight of his other eye because of cataracts.

Euler’s abilities were exceptional. It was said that ‘Euler calculated without effort, as men breathe’. He could work in a noisy environment and often wrote at home while his children played around him. Euler had thirteen children, although only five lived to adulthood. He was able to go on working even after he became blind because his memory was so good that he could do complex calculations in his head.

Euler made significant contributions to almost every area of mathematics including geometry, calculus, mechanics, and number theory. He was one of the first to make use of standard symbols including π for Pi, Σ for summation (adding), f (x) for functions, ex for the natural logarithms, and i for complex numbers. He also worked on a proof of Fermat’s last theorem and was able to solve it for powers of 3 and 4. Like most mathematicians, he worked on finding formulas for the value of π. He probably wrote more books than any other mathematician in history. In his lifetime he published more than 500 books and papers, and another 400 were published after his death.

Apparently, his last words were simply ‘I die’.

Questions1. What did Euler lose when he was 31

years old?2. What ability did Euler have that

allowed him to go on working when he became blind?

3. Name four areas of mathematics in which Euler made significant contributions.

4. How many books and papers did Euler publish during his life?



Basic properties of networks

1 State the degree of each vertex in the network shown at right.

2 State the degree of each vertex for the networks shown below.a b

c d

3 Find the number of regions, edges and vertices for each of the networks in question 2.

4 a Find the number of vertices, edges and regions in the network shown at right.

b Confirm Euler’s formula for the network shown at right.

remember1. The degree of a vertex is the total number of edges to which that vertex is

connected.2. The total of all the degrees of a network is exactly twice the total number of

edges.3. A planar network is a network where edges do not cross each other.4. A planar network has faces (or regions).5. In any planar network, if E = the number of edges, F = the number of faces or

regions and V = the number of vertices, then: V = E − F + 2.This is called Euler’s formula.

remember

13BD

E F

A

B

CWWORKEDORKEDEExamplexample

4

R

V T

S

QA

C

F E

DB

G

P Q

R S

TU

V W

E

D

F B

A

C


5

Mathcad

Euler’srule

D

E F

A

B

C


6



5 Confirm Euler’s formula for each of the networks in question 2.

6 Consider the network defined by the following list of edges.{(A, B), (A, C), (B, E), (B, F), (C, D), (C, E), (D, E), (E, F)}

a Construct a drawing of the network (make sure it is planar).

b Determine the degree of each vertex.

c Confirm Euler’s formula for this network.

7A planar network consists of:

8The sum of the degrees of all the vertices of a network equals:

A regions, faces and edgesB regions, vertices and nodesC faces, edges and verticesD edges which crossE none of the above

A an odd numberB half the number of verticesC twice the number of verticesD half the number of edgesE twice the number of edges

GAME

time

Networks — 001


WorkS

HEET13.1


MA

TH

SQUEST

C H A L L

EN

GE

MA

TH

SQUEST

C H A L L

EN

GE

1 This cube is made of wire. Each edge of the cube is 1 metre long. An antstarts at corner A and walks along as many edges as possible withoutgoing along the same edge twice. How far does the ant walk?

2 The cube shown in part 1 is made from a length of wire by bending,cutting and soldering joints only when necessary. There must be noloose end of wire. What is the least number of joints to be soldered?

A



1 Draw a network, which represents the following family tree.Ann and Charlie marry and they have 2 children, Bob and Karen.Bob marries Delia and they have a single child, Peter.Peter marries Mary and they have 2 children, Ben and Emily.

2 Four streets are connected to each other as follows. Draw a network represented bythese connections.Princess Street–Bird AvenuePrincess Street–Charlie StreetCharlie Street–Dundas StreetPrincess Street–Dundas Street

3A network consists of edges and:A letters B sides C pictures D shapes E vertices.

4 Draw the network represented by the following list.{(A, B), (A, C), (A, D), (B, G), (C, D), (C, E), (C, F), (D, E), (E, F), (F, G)}

Use the network shown to answer questions 5 and 6.

5 Represent the network shown at right usingthe labelling method.

6 Find the number of vertices, edges andregions in the network at right.

Use the following network for questions 7 and 8.

7 What is the degree of vertex B?

8 Find the number of vertices (V), edges (E)and faces (F) in the network shown andconfirm Euler’s formula.

The following list of words are to be used to fill in the missing words for questions 9and 10.

Planar, vertices, edges, degrees, objects, network, twice

9 A network consists of faces, edges and .

10 The sum of the of all the vertices of the network equals

the number of .

1


A

C

B

D E

A

D

F

C

E

B



Application of networks to problem-solving

The handshake problemProblems involving contact between people, for example how many handshakes orgreetings are needed to include everyone, can be solved using networks, by followingthe steps outlined in the next worked example.

Consider a group of 6 people sitting at a round table. How many handshakes are required so that each person shakes hands with everyone exactly once?

THINK WRITE/DRAW

Draw a circle representing the table and the 6 vertices, A–F, representing the 6 people at the table.

Start with person A. This person must shake hands with 5 other people — B, C, D, E and F. Show this by adding the appropriate connections to the diagram (red lines).

1 A

D

F B

E C

2 A

D

F B

E C

7WORKEDExample



A network where every vertex is connected to each other vertex is called a completenetwork. The final network in the handshake problem above is an example of acomplete network.

TournamentsThe set of steps used to find the number of handshakes considered in the previousworked example can also be used to work out games or teams in a ‘round-robin’tournament, where each player or team gets to play all the others exactly once. Theworked example which follows shows a different approach to the problem for 6teams.

THINK WRITE/DRAW

Now, consider person B. This person has already shaken hands with A (see step ) and still needs to shake hands with the remaining 4 people — C, D, E and F. Add 4 connections to the diagram to show the handshakes of person B (blue lines).

For person C, the three people remaining to shake hands with are D, E and F. Add appropriate edges to the diagram (black lines).

Person D still needs to shake hands with two people — E and F.Show this with edges DE and DF (green lines).

The last handshake is between E and F (orange line).

We now count the total number of handshakes.

The total number of handshakes= 5 + 4 + 3 + 2 + 1 = 15

3

2

A

D

F B

E C

4 A

D

F B

E C

5 A

D

F B

E C

6 A

D

F B

E C

7



8WORKEDExampleSix waterpolo teams (A to F) are playing in a knockout tournament. Use networks to organise the games so that each team plays every other team once.

THINK WRITE/DRAW

Begin by making a ‘ring’ joining each vertex to its nearest neighbour.

Round 1Starting at A and, going clockwise, each team plays the team on its left as shown with red lines.

Round 2Starting at A and, going counter-clockwise, each team plays the team on its right as shown with blue lines.

Games played

Round 1:A plays BC plays DE plays F

Round 2:A plays FE plays DC plays B

Round 3Join A to the ‘opposite’ vertex and then make connections at right angles to this connection (blue lines).

Round 3:A plays DB plays FC plays E

Round 4Join B to the ‘opposite’ vertex and then make connections at right angles to this connection (black lines).

Round 4:A plays CB plays ED plays F

Round 5Join C to the ‘opposite’ vertex and then make connections at right angles to this connection (green lines).

Round 5:A plays EB plays DC plays F

1 A

D

F B

E C

2 A

D

F B

E C

3 A

D

F B

E C

4 A

D

F B

E C

A

D

F B

E C


C h a p t e r 1 3 N e t w o r k s 547By observing the listed games in each round, you can confirm that each team doesindeed play each other team exactly once.Note: The number of games played by the 6 teams is equal to the total number of hand-shakes between 6 people.

Odd numbers of teamsA further difficulty arises when an odd number of teams play. One very elegant solutionis to add a ‘dummy’ team, so that if there are originally 5 teams, then the dummy is thesixth team. Whoever is scheduled to play the dummy actually gets a ‘bye’ for thatround. Thus, you could use the method in worked example 8 for a tournament of5 teams, with F as the dummy and therefore there would be byes for teams E, A, B, Dand C in the 5 rounds.

Application of networks to problem-solving

1 Solve the handshake problem for 5, 7, 8, 9 and 10 people and hence, complete the following table.

2 Use networks to determine theschedule for a tournament inwhich the following numbers ofsoccer teams play.a 4b 7c 8d 9e 10

Number of people 5 6 7 8 9 10

Number of handshakes 15

remember1. A network where every vertex is connected to each other vertex is called a

complete network.2. We aim to construct a complete network when solving problems where each

person (or team) needs to contact (or play) every other person (or team).3. If the number of teams in a tournament problem is odd, add a ‘dummy’ team

and proceed as you would with an even number of teams.

remember

13C

EXCEL Spreadsheet

Handshakecounter


7


8



Paths and circuits — part I As we have seen, a network can represent manyphysical situations, such as roads andtowns, computers and cables, peopleand handshakes or even classroomsand doorways. There are certainlysituations where each edge andeach vertex is equally important.With planar graphs we candefine a sequence of edges andvertices and we can look atvarious sequences through thenetwork.

Sometimes you may wish tohave a sequence that goesthrough all vertices only once;for example a travelling sales-person who wishes to visit eachtown or shop once.

Sometimes you may wish to use alledges only once, such as a road repair gangrepairing all the roads in a shire.

Both these cases are known as traversable networks.

Paths and circuitsA path is a sequence of edges and vertices that begins at one vertex and ends at another. A circuit is a sequence which begins and ends at the same vertex.

Consider the network shown on the right. One path, goingthrough each vertex is E–D–C–B–A. The vertices might rep-resent 5 different tourist attractions you would like to visit inone day during your holidays, and the edges might show theroads connecting those attractions.

One circuit would be E–D–C–B–A–E. In this case, E mightrepresent your starting and ending point (say, your hotel) and the other 4 vertices maybe the 4 tourist attractions you would like to visit.

Euler paths and circuitsAn Euler path (and circuit) uses each edge exactly once.

Euler paths and circuits are useful when the edges are the really important objects, suchas a network of roads connecting towns. This would be the case if you were designinga bus route which travels along each road, picking up and dropping off passengers.Unfortunately, not all networks have Euler paths or circuits.

What determines whether or not there is an Euler path or circuit? The answer lies indetermining the degree of each vertex and whether each degree is even or odd.

E

DA

BC


C h a p t e r 1 3 N e t w o r k s 549The following table shows the presence or absence of Euler paths and circuits for

networks with different numbers of odd degree vertices.

The following worked example provides more guidance in locating an Euler path.

Degree of vertices Euler path How? Euler circuit How?

All the vertices have even degrees.

Yes Start and finish at any vertex.

Yes This depends upon the actual network.

There are exactly two odd degree vertices and the rest are even.

Yes Start and finish at the two odd degree vertices.

No

There are any other number of odd degree vertices (1, 3, 5, and so on).

No No

Determine if there is an Euler path through the network shown at right and, if so, give an example.

THINK WRITE/DRAW

Determine the degree of each vertex by counting the number of edges connected to it.

Vertex A has a degree = 2.Vertex B has a degree = 2.Vertex C has a degree = 3.Vertex D has a degree = 2.Vertex E has a degree = 3.

Count the number of odd degree vertices and hence state whether there is an Euler path through the network.

Number of odd vertices = 2Therefore, the Euler path exists.

Since there are exactly 2 vertices with odd degrees (C and E), an Euler path has to start and finish with these; say, begin at C and end at E. Attempt to find a path which uses each edge.Note: While each edge must be used exactly once, vertices may be used more than once.To ensure that each edge has been used, label them as you go.

List the sequence of vertices along the path. An Euler path is C–B–A–E–C–D–E.

1

2

3 E

DA

BC

6 3

2

1

4

5

4

9WORKEDExampleE

DA

BC



Note: The number of vertices in the path is one more than the number of edges in thenetwork. For instance, in worked example 9 there were 6 edges in the network and thepath contained 7 vertices; that is, one more vertex than the total number of edges.

Can you find a different sequence for the network in the previous worked example,which is also an Euler path? Does this network have an Euler circuit? Why (or whynot)?

Hamiltonian paths and circuitsA path which goes though each vertex exactly once is known as Hamiltonian. If a Hamiltonian path begins and ends at the same vertex, it is then a Hamiltonian circuit.

Unfortunately, there is no set method of finding these routes, except by inspection ortrial and error. Most networks have both Hamiltonian paths and circuits. Finding acircuit may require a clever choice of the starting (and ending) vertex.

Hamiltonian paths and circuits are most useful in ‘travelling salesperson’ types of prob-lems where the vertices might represent places that the salesperson must visit each week.

10WORKEDExampleFor the network shown find:a a Hamiltonian pathb a Hamiltonian circuit.

THINK WRITE

a Choose any starting vertex and attempt to visit all the other vertices. (Not all the edges need to be used.)

a A possible Hamiltonian path isA–B–F–E–D–C.

b Attempt to ‘close’ the path by returning to the starting vertex.

b A possible Hamiltonian circuit is A–B–F–E–D–C–A.

FB

A

D

C E

remember1. A path is a specific set of edges and connecting vertices.2. An Euler path is a set of edges connecting all the vertices such that each edge

is used exactly once. For such a path to exist, there must be exactly two vertices of odd degree (and the rest of even degree) or all the vertices must be of even degree.

3. An Euler circuit is an Euler path that also has the same starting and ending vertex. For such a circuit to exist, all the vertices must be of even degree.

4. A Hamiltonian path travels through each vertex exactly once.5. A Hamiltonian circuit is a Hamiltonian path that also has the same starting and

ending vertex.

remember



Paths and circuits — part I

1 Determine if there is an Euler path through the network shown at right and, if so, give an example.

2 Determine an Euler path through the networks shown below.a b c

3 Starting at vertex R, determine an Euler path for this planar graph.(Hint: What vertex should the path end at?)

4 A road inspection crew must travel along eachroad, shown on the map at right, exactly once.a From either of which two cities must the crew

begin its tour? b Determine a path using each road once.c Which cities are visited most often and why?

5 For each of the networks shown, find:i a Hamiltonian pathii a Hamiltonian circuit.a b c d

6 Which of the networks in the four parts of question 5 have Euler paths?

7An Euler path:

8A Hamiltonian path:

A uses every vertex exactly once B uses every edge exactly onceC always exists D is the same as a Hamiltonian pathE always starts on a vertex whose degree is an even number

A does not exist if there are any vertices of odd degree

B does not exist if there are exactly2 vertices of odd degree

C does not have to use every vertex in the network

D does not have to use every edge in the network

E is the same as a Euler path

13DA B

C

E D


9

A B

C

E D

Q S

R T

P U

Q S

R T

P U

Q S

R T

P U

Alhambra

HaplessBuford

Chesterton Grunge City

French Twist

Eulersburg

DullsvilleWWORKEDORKEDEExamplexample

10

A B

C

E

D

A B

C

F

G E

D

A B

C

F

G E

D

B D

A F

C

E





Paths and circuits — part II As was mentioned in the previous section, finding a path which starts and ends at thesame vertex and uses all the edges exactly once (an Euler circuit) can be difficult. Inthis section we will explore techniques to help us find Euler circuits.

Finding an Euler circuitRemember that an Euler circuit exists only if all the vertices are of even degree.

Which method we use to find an Euler circuit depends upon the starting vertex wechoose. For instance, if the starting vertex in the above example was B, it would bepointless to select the shortest path (B–A–D–B) back to it, as it would be a dead end.An Euler circuit from B is B–A–C–E–A–D–B.

Generally, if the starting point has the degree of 2, the first and the last connectionsof the circuit must be to that point. This means that the shortest path back to the startingpoint is no use, as there are no more edges from it and it would be a dead end. Insteadwe can first connect the starting vertex with the vertex of larger degree, as shown in thefollowing example.

Starting with vertex A, find, if possible, an Euler circuit for the network shown.

THINK WRITE/DRAW

Confirm that all of the vertices are of even degree.

Vertex A has a degree = 4.Vertex B has a degree = 2.Vertex C has a degree = 2.Vertex D has a degree = 2.Vertex E has a degree = 2.All vertices are of even degree, therefore an Euler circuit is possible.

Start with vertex A and find the shortest path back to vertex A. (If there is more than one such path, select any one of them.) A possible shortest path is A–C–E–A.Complete the circuit by going around the remaining vertices (along pathA–D–B–A).

Confirm that each edge is used once, and list the path.

The Euler circuit is A–C–E–A–D–B–A.

1

2

B

A

D

C

E

12

3

3

B

A

D

C

E

1

4

5

6

2

3

4

11WORKEDExample

B

A

D

C

E



Note: We had to have the path CD as the last one, since we chose DA as the first.

Find an Euler circuit for the network shown.

THINK WRITE/DRAW

Confirm that all of the vertices are of even degree.

Vertex A has a degree = 4.Vertex B has a degree = 2.Vertex C has a degree = 4.Vertex D has a degree = 2.Vertex E has a degree = 2.All vertices are of even degree, therefore an Euler circuit is possible.

Choose a starting vertex, say D. Since D is of degree 2, the first and last connections must be to D. Therefore, we can not look for the shortest path back to D, as it would create a dead end. Instead, we need to first connect D to the vertex of larger degree. There are 2 such vertices, A and C (both of a degree 4). So, connect D to one of them, say, A.Find a ‘shortest path’ back to A (select any, if there is more than one such path).

Complete the circuit by moving from A back to D, using the remaining edges.

List the path. The Euler circuit is:D–A–B–C–A–E–C–D.

1

2A

D E

CB1

3

A

D E

CB1

2 3

4

4

A

D E

CB1

2 3

4

5

76

5

12WORKEDExample

A

D E

CB

remember1. For an Euler circuit to exist, all the vertices must be of even degree.2. Complete a circuit by finding the ‘shortest’ path to the starting vertex, adding

new vertices one at a time until all vertices are part of the circuit.3. If the starting vertex has the degree of 2, connect it to a vertex of larger degree first.

remember



Paths and circuits — part II

1 Starting with vertex A, find, if possible, an Euler circuit for the network shown.

2 Starting with vertex D, find an Euler circuit for the network in question 1.

3 Starting with vertex A, find an Euler circuit for the following networks.a b

4 Find an Euler circuit for the networks shown.a b c

5 In a computer network, the main computer, or file server (F), isconnected to all the other computers as shown at right. Matsuko,the technician, wishes to test that each connecting cable is func-tioning properly. She wishes to route a signal, starting at F, sothat it travels down each cable exactly once and then returns tothe file server. Find such a circuit for the network configurationshown at right.

13EA

D

B

E C


11

A

F

CB

E D

B DC

A F

E


12

B

F G

D

A

E

CBA

E

F

C D

B C

E

F

G

DA

F

W

P

R

ST

UV

Q



If a travelling salesman wished to visit each town connected by a network of roadsexactly once and then return home, he would be looking for:

7A group of towns are connected by railroad, as shown in the simpli-fied network at right. The shire wishes to be able to have a path sothat each rail link can be used exactly once. Where should it add anew link?

A an Euler path B an Euler circuit C a Hamiltonian pathD a Hamiltonian circuit E none of the above

A Between J and M B Between L and NC Between P and M D Between K and ME Between P and L

SproutsSprouts is a simple pencil and paper game for two players. Here are the rules:

1 Players take turns at being the first to start play.

2 The first player marks 3 separate dots on a piece of paper.

3 The second player draws a line joining two of the dots and adds a dot somewhere on the line just drawn.

4 The two players continue to take turns at drawing a line joining two of the dots and adding a new dot somewhere on this line.

However, there are some conditions!

• No dot is allowed to have more than three lines leaving it. (A dot is ‘dead’ when three lines are leaving it and can be circled to indicate this.)

• No new line may touch or cross any line already drawn.

5 The winner is the last person able to play.

Keep a record of your games and try to develop a winning strategy. You can also play this game with a different number of starting dots.


WorkS

HEET13.2mmultiple choiceultiple choice

J

K L

P

M

N



1 Represent the network shown at right by using the labelling method.

2 Draw the network represented by the following list.{(A, B), (A, C), (A, D), (A, E), (B, C), (C, D), (C, E)}

Use the network shown at right for questions 3 and 4.

3 Find the number of vertices (V), edges (E) and regions (R) in the network shown.

4 Confirm Euler’s formula for the network shown.

5 Determine an Euler path through the network shown at right.

6For an Euler path to exist there must be exactly 2 vertices of odd degree or:A all vertices must be of odd degreeB one vertex is of odd degreeC one vertex is of even degreeD there is an equal number of odd and even degree verticesE all vertices must be of even degree.

Use this network for questions 7 and 8.

7 Find a Hamiltonian path.

8 Find a Hamiltonian circuit.

Answer true or false for questions 9 and 10.

9 An Euler path uses every edge only once.

10 A Hamiltonian circuit is a Hamiltonian path that does not have to have the samestarting and ending vertex.

The bridges of KönigsbergDo you remember the problem posed at the start of the chapter (see page 531)? Can it be done? Can you travel across all 7 bridges of Königsberg once, and only once, before returning to the starting point? 1 Try this problem again. (The university students of Königsberg actually called

in Leonhard Euler in 1735 to help them with this problem.)2 Explain your answer in terms of networks and odd and even vertices.3 Draw a diagram and add an eighth bridge so that you could travel over all the

bridges once only if you started and finished at different points.4 Draw another diagram. Can you add an eighth bridge so that you could travel

over all the bridges once only if you started and finished at the same point? Hint: Each bridge can be represented by an edge and the islands and riverbanks represented by vertices, as shown at left.

2A D F

B C E

C

D

E

A

FB

E

A

B D

C


A G

F

B

ED



Trees and networks A special kind of network is called a tree. Like woody trees network trees have branches.In network trees, the branches, called edges, come together at points, called vertices.

A tree consists of a number of vertices and the smallest number of edges necessary, so that each vertex is connected to at least one other vertex.

This ensures that each vertex can ‘communicate’ to all the other vertices either directly,or indirectly.

Identifying treesThe diagrams on the right show trees for 2 and3 vertices respectively. For the 3-vertex case,there are 2 other possible trees. Can you drawthem?

The next two diagrams show trees for 4 and 5vertices respectively. Observe that for 4 verticesthere are 3 edges and for 5 vertices there are 4edges.

Generally, a tree has one less edge than the number of vertices.

The last figure at right is an example of a network, which is nota tree. Note that it has one too many edges. Can you identify theextra edge (that is, the edge that has to be removed in order for thenetwork to become a tree)?

A B A B

C

A B

C

D

A B

C

D

E

A B

C

D

E



Identify which of the following networks are trees.a b c

d e

THINK WRITE

a Count the number of vertices and edges. There should be one edge less than the number of vertices.

a Number of vertices = 6Number of edges = 5

Check whether each vertex is connected to at least one other and state your conclusion.

A is connected to B.B is connected to F.C is connected to D.D is connected to E.F is connected to B.Therefore, the network shown is a tree.

b Count the number of vertices and edges. The number of edges must be one less than the number of vertices for the network to be a tree.

b Number of vertices = 6Number of edges = 6Therefore, the network shown is not a tree.

c Count the number of vertices and edges. There should be one edge less than the number of vertices.

c Number of vertices = 7Number of edges = 6

Check whether each vertex is connected to at least one other and state your conclusion.

A is not connected to any vertex.Therefore, the network shown is not a tree.

d Check whether the number of edges is one less than the number of vertices.

d Number of vertices = 7Number of edges = 6

Ensure that each vertex is connected to at least one other.

Each vertex is connected.Therefore, the network shown is a tree.

e Check if the number of vertices exceeds the number of edges by one.

e Number of vertices = 7Number of edges = 6

Ensure that each vertex is connected to at least one other.

Each vertex is connected.Therefore, the network shown is a tree.

A B

C

D

F

E

A B

C

D

F

E

A G

D

B

C

F

E

A G

D

B

C

F

E

R S

T

U

Q

P

V

1

2

1

2

1

2

1

2

13WORKEDExample



The minimal spanning treeConsider a network of vertices and edges where the edges represent, say, the distancebetween the vertices. We wish to find a tree within the network, so that the total dis-tance along the tree is as small as possible.

This is called the minimal spanning tree and is useful, for example, for a messagesystem, where the total distance travelled between the vertices should be as small aspossible. The method for finding the minimal spanning tree is particularly simple, andis demonstrated in the next worked example.

In the network shown, numbers represent the distances (in km) between ten towns. Find the minimal spanning tree and determine its total length.

Continued over page

THINK WRITE/DRAW

Find the edge with the shortest distance. It is the start of the tree. The shortest edge is T–V.

Total distance = 10

Find the edge connecting a new vertex to the existing tree (T–V) by the shortest distance.Town T is connected to S (distance 27 km), Q (23 km) and U (31 km). Town V is connected to W (24 km), X (26 km) and U (25 km).Choose the smallest distance, T–Q (23 km) and find the total distance so far.

Total distance = 10 + 23= 33

1R S

Q

P

W

XT V

U Y

15 33

23 10 26

31 252021

35 27

3818

3424

2R S

Q

P

W

XT V

U Y

15 33

23 10 26

31 252021

35 27

3818

3424

14WORKEDExample

R S

Q

P

W

XT V

U Y

15 33

23 10 26

31 2520

21

35 27

3818

3424



THINK WRITE/DRAW

Find the edge connecting a new vertex to the existing tree (Q–T–V) by the shortest distance:Town Q is connected to R (35 km),P (21 km) and U (20 km).Town T is connected to S (27 km) and U (31 km).Town V is connected to W (24 km), X (26 km) and U (25 km).

Choose Q–U (20 km) and find the new total distance.


Find the edge connecting a new vertex to the existing tree (U–Q–T–V) by the shortest distance. Using the same method as in previous steps, selectQ–P (21 km).

Calculate the new total. Total distance = 53 + 21= 74

The edge connecting a new vertex to the existing tree (U–P–Q–T–V) by the shortest distance is V–W (24 km).

Add 24 to the previous total. Total distance = 74 + 24= 98

The edge with the shortest distance connecting a new vertex to the existing tree (U–P–Q–T–V–W) is V–X (26 km).Note: If U–V was used the figure would no longer be a tree.


3R S

Q

P

W

XT V

U Y

15 33

23 10 26

31 252021

35 27

3818

3424

4R S

Q

P

W

XT V

U Y

15 33

23 10 26

31 252021

35 27

3818

3424

5R S

Q

P

W

XT V

U Y

15 33

23 10 26

31 252021

35 27

3818

3424

6R S

Q

P

W

XT V

U Y

15 33

23 10 26

31 252021

35 27

3818

3424



It is now possible for each vertex in worked example 14 to ‘communicate’ with all theothers, with a minimum total distance of 184 km. Suppose there was a major fire intown T. They would send messages directly to Q, S, and V. These towns would send amessage to towns P, U (from Q), R (from S), W and X (from V) and so on, until everytown had received the message.

Although finding the minimal spanning tree may seem a long process, at each stepthe decisions are very simple: just add the shortest edge to a new vertex.

THINK WRITE/DRAW

The next edge with the shortest distance connecting a new vertex to the existing tree (U–P–Q–T–V–W–X) is X–Y (18 km).


Select edge T–S (27 km). It connects a new vertex (S) to the existing tree (U–P–Q–T–V–W–X–Y) using the shortest distance.


Finally, select the edge S–R (15 km), which connects the last remaining vertex, R, to the existing tree (U–P–Q–T–S–V–W–X–Y) and has the shortest distance.The tree is now complete, as all vertices are connected.Calculate the total and answer the question.


Total length of the minimal spanning tree is 184 km.

7R S

Q

P

W

XT V

U Y

15 33

23 10 26

31 252021

35 27

3818

3424

8R S

Q

P

W

XT V

U Y

15 33

23 10 26

31 252021

35 27

3818

3424

9R S

Q

P

W

XT V

U Y

15 33

23 10 26

31 252021

35 27

3818

3424

remember1. A tree uses the smallest number of edges possible, so that each vertex is

connected to at least one other vertex. 2. A tree of n vertices contains (n − 1) edges. (That is, the number of edges is one

less than the number of the vertices.)3. The minimal spanning tree is the tree that has the shortest total overall length.4. To find the minimal spanning tree, start with the shortest overall edge. Add a

new vertex by finding the shortest edge connecting it to the existing tree, until all vertices are connected.

remember



Trees and networks

1 Identify which of the following networks are trees. a b c d e

2 In the network at right, identify one (or more) edges which,when removed, result in the remaining network becoming atree.

3 Identify one (or more) edges in the figure at right which, whenadded, result in the network becoming a tree.

4 In the network shown at right, numbers represent the dis-tances (in km) between seven towns. Find the minimalspanning tree and determine its total length.

5 The diagram on the right represents 8 cities and the roadsconnecting them. The distances along each road are alsoindicated. If the distances are in km, find the minimalspanning tree for this network. Determine the total lengthof the tree.

13FWWORKEDORKEDEExamplexample

13A

C

D

BE

BF

CA

D C

A F

B D

E

C

L

J M

NK

D A I E

H G C

F B

A B

E D C

F

U WZ

Y

X

T V

A C

D E

F

GB51

67

48

39

34

4137

464536

44


14

Q

IY

W E

TR

O

61

7762

45 6870

77

55 52 72

6181

67


C h a p t e r 1 3 N e t w o r k s 5636 Two new towns (X and Z) and their road connections

are added to the network in question 5, as shown. Findthe new minimal spanning tree and its length.

7 A group of computers areconnected as shown below.The numbers on the edgesrepresent the cost (in cents) ofsending a 1 MB mail messagebetween the computers. Findthe smallest possible cost tosend a message to all the com-puters.

8A Hamiltonian circuit is not a tree because:

9The next edge that should be added to the tree shownby red lines in the figure at right, to form a minimalspanning tree, would be:

A not all the vertices are connected in a Hamiltonian circuitB there are more vertices than edges in a Hamiltonian circuitC there are the same number of vertices and edges in a Hamiltonian circuitD there are more edges in a Hamiltonian circuit than are required for a treeE a Hamiltonian circuit does not necessarily connect all the vertices.

A A–D (length = 15)B C–E (length = 16)C B–C (length = 18)D A–B (length = 19)E G–F (length = 23)

Q

I

X

Y

Z

W E

TR

O

61

7762

45 6870

77

5552 72

6181

4936

67

40

44

36

26

14 19

21

24

40

1118

23

3216

1612

1727

GAMEtime

Networks— 002


WorkS

HEET13.3mmultiple choiceultiple choice B

A C

D E

G F

19 18

159 16

11

12

1320 22

23



Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows.

1 A network consists of edges and .

2 When using the labelling method to describe a network, the pair { . . .(C, F) . . . } means that C is to F.

3 The degree of a vertex is the total number of it has con-nected to it.

4 In a planar graph there are no edges.

5 Euler’s rule connects the number of edges, vertices and witha formula V = E − F + 2.

6 If there are 4 people in a room, the smallest number of handshakes, so thateach person shakes hands with everyone else exactly once is .

7 A is a sequence of edges and vertices that begins at onevertex and ends at another, while a is a sequence whichbegins and ends at the same vertex.

8 An Euler path uses each exactly once.

9 An Euler path is possible, if either vertices in the network areof even degree, or vertices are of odd degree.

10 A uses each vertex exactly once.

11 A Hamiltonian circuit does not necessarily use in the network.

12 A network with two vertices of degree equal to 3 cannot have an.

13 A tree consists of a number of vertices and edges necessary,so that each vertex is connected to at least one other.

14 A tree for a network with 12 vertices would have edges.

15 A minimal spanning tree has possible.

summary

W O R D L I S TEuler circuit11verticesthe smallest total

lengthedges

regionsevery edge2circuitedgeall

Hamiltonian pathvertexthe smallest

number of

connectedcrossing6path



1 Draw the following network.{(A, B), (A, C), (A, D), (B, E), (C, E), (C, F), (D, F), (E, G), (F, G)}

2 Represent the network shown, using the labelling method.

3 State the degree of each vertex in the network in question 2.

4

For the network shown above:a Count the number of

vertices (V).b Count the number of

edges (E).c Count the number of

faces (F).d Confirm Euler’s formula.

5 Eight beach volleyball teams (A to H) are playing in a knockout tournament. Use networks to organise the games so that each team plays every other team once.

13A

CHAPTERreview

13AQ

T U

SR

V

A X Y B

W

13B13BA F

C

E

B D

13C



6 How many handshakes will it take if there are 8 people and everyone shakes hands with everyone else exactly once?

7 For each of the following networks, state whether an Euler path and/or an Euler circuit are/is possible.a a network with 2 vertices with the degree = 2 and 2 vertices with the degree = 3b a network with 3 vertices with the degree = 2 and 2 vertices with the degree = 3c a network with 3 vertices with the degree = 2 and 3 vertices with the degree = 3d a network with 2 vertices with the degree = 2 and 3 vertices with the degree = 3e a network with 5 vertices with even degreesf a network with 4 vertices with odd degreesg a network with 1 vertex with the degree = 2, 2 vertices with the degree = 3 and 2 vertices

with the degree = 4

8

a For the network shown above, find:i an Euler path ii a Hamiltonian path iii a Hamiltonian circuit.

b Explain why is there no Euler circuit.

9 Modify the network in question 8 so that there is a possible Euler circuit. Draw a circuit, starting with vertex A.

10 a For the network shown at right, state two possible Euler paths, starting with different vertices.

b Modify the network by adding a new vertex, F, so that an Euler circuit is possible.

c For the modified version of the network, state an Euler circuit, beginning with:i vertex Bii vertex C.

11 How many edges are required for a tree, connecting:a 4 vertices? b 8 vertices? c 100 vertices?

12

Find the minimal spanning tree for the network shown and determine its total length.

13C

13D

13D A F

C

E

B D

13E

13E

A E

B

C

D

13F

13F VR

X

SY

P

T

Z

Q

W

U

30 29

15

3621

17

318

1827

26 23

2520

11 1033

20

3432

testtest

CHAPTERyyourselfourself

testyyourselfourself

13


network math

Documents