Undirected graphs and networks

Undirected
graphs and
networks
21
VCE VCEco coverage erage
Area of study
Units 1 & 2 • Geometry
In this chapter
21A Vertices and edges
21B Planar graphs
21C Eulerian paths and
circuits
21D Hamiltonian paths and
circuits
21E Trees

154 General Mathematics
Undirected graphs and networks
Graphs are an efficient way of summarising data in many practical problems. The
graphs we will be dealing with in this chapter differ from those that we have worked
with in the past, as they consist of points connected by various lines. As there is no
particular order or direction to these lines, the graphs are defined as undirected graphs
or networks.
Undirected graphing is an area of mathematics dealing with problems such as planning
a delivery route to visit a number of shops while travelling the least distance,
designing a communications network to link a number of towns, organising the flow of
work in a factory, or allocating jobs for increased efficiency.
Below are examples of undirected graphs or networks you may have come across:
Route map for the Melbourne Metropolitan Tram Network
H – O – H O= C = O
(H 2 O) (CO 2 )
The chemical molecules for water (left)
and carbon dioxide (right)
Milliammeter
Rectifier
OA91
12 v, 24 w Lamp
1000 ohm
Voltmeter
+
The Swiss mathematician Leonhard Euler (pronounced oyler; 1707–1783) developed
much of the theory of undirected graphs in his work on topology and graph theory.
AC
50 mA
+
DC
+ –
An electrical circuit
+

Chapter 21 Undirected graphs and networks 155
Vertices and edges
The map at right shows the main roads linking a number of
country towns in Victoria.
We can see that there are three main roads leading into
Traralgon, but only one joining Leongatha and Yarram. The
map is an example of an undirected graph or network since
there are no arrows showing a particular direction.
The network consists of vertices (the towns) and edges (the roads). The degree of a
vertex is defined as the number of edges leading to or from that vertex. Therefore the
degree of the Traralgon vertex is 3 because there are 3 roads leading to or from there. The
degree of the Yarram vertex is 2 because there are only 2 roads leading to or from there.
The graph is considered to be a connected graph since it is possible to reach each
vertex from any other one. A connected graph must have all vertices joined to at least
one other vertex. There cannot be any isolated vertices.
We use the term multiple edge if there is more than one edge
linking two vertices. For example, in the figure at right there is a
multiple edge between vertices A and C.
The figure also contains a loop at vertex B. An edge which
connects a vertex to itself is defined as a loop. When calculating
the degree of a vertex, a loop counts as 2. Thus the degree of vertex
B is 3; 1 for the edge connecting B to A plus 2 for the loop.
Trafalgar Moe
Morwell
Traralgon
Leongatha
Yarram
The figure is not a connected graph because not all vertices are connected to at least
one other vertex. E is therefore an isolated vertex as it is not able to be reached from
the other vertices.
WORKED Example
For the following graph, state:
a the number of vertices b the number of edges
c the degree of each vertex d whether the graph is connected.
THINK WRITE
a Count the number of vertices.
Note: The vertices are the points labelled
A, B, C, D and E.
1
b Count the number of edges.
Note: The number of edges is found by
counting the number of lines joining the
vertices. Remember: A loop is counted as 2.
c Count the degree of each vertex, that is, the
number of edges leading to or from each
vertex.
Note:
1. C contains a loop, which is counted as 2
when totalling edges.
2. The degree of A is abbreviated to deg(A)
etc.
a The number of vertices is 5, that is,
A, B, C, D and E.
b The number of edges is 9.
c Vertex A has 4 connections to
other vertices, so deg(A) = 4.
Vertex B has 3 connections to
other vertices, so deg(B) = 3.
Vertex C has 6 connections to
other vertices, so deg(C) = 6.
Vertex D has 3 connections to
other vertices, so deg(D) = 3.
Vertex E has 0 connections to
other vertices, so deg(E) = 0.
d Answer the question. d The graph is not connected, as vertex E has
no edges leading to or from it; that is, it is
isolated from each of the other vertices.
A
D
A
C
D
B
E
A
B
B
C
C
E
E
D

156 General Mathematics
WORKED Example
2
Draw a connected graph which has 6 vertices, 2 loops and 3 multiple edges. Determine the
number of edges and the degree of each vertex.
Note: There are numerous ways of drawing this connected graph using the given information.
THINK WRITE
1 Draw 6 points and label them
A, B, C, D, E, F.
2 Draw edges connecting the points A to F.
Note: As this diagram represents a connected
graph, each vertex must be connected to at least
one other vertex.
A B C
3 Select two of the points and draw a loop on
each of them.
D
4 Select 3 pairs of points and draw an extra edge
connecting each specific pair.
Note: For this example, each step has been drawn
in a different colour to highlight each stage.
E F
5 Count the number of edges.
Note: A loop is counted as 2 when totalling
edges.
There are 13 edges.
6 Determine the degree of each of the vertices,
that is, count the number of edges leading to
deg(A) = 3; that is, A Fig 1
or from each vertex.
Note: Vertices B and C have loops; thus they
will each have an extra 2 added to their vertex
deg(B) = 5; that is, Fig B 14
sum.
deg(C) = 3; that is, C
remember
remember
deg(D) = 4; that is,
deg(E) = 3; that is,
deg(F) = 4; that is,
D
F
E
Fig 17
1. An undirected graph or network consists of vertices and edges.
2. The degree of a vertex equals the number of edges connected to it.
3. A loop adds 2 to the degree of a vertex.
4. A connected graph has no isolated vertices.

WORKED
Example
1
WORKED
Example
2
21A
Chapter 21 Undirected graphs and networks 157
Vertices and edges
1 For each of the following graphs, state:
a A
B
b
A
B c
D
d e f
A B
A
C D
D
i the number of vertices
ii the number of edges
iii the degree of each vertex.
C
C
2 Which of the graphs in question 1 are connected?
3 Draw the graphs in question 1 that are not connected, and include extra edges to make
the graphs connected.
4 Which of the graphs in question 1 contain loops?
5 Draw a connected graph which has 7 vertices, 3 loops and 4 multiple edges. Determine
the number of edges and the degree of each vertex.
6 Draw a connected graph that has:
a 5 vertices and 8 edges b 5 vertices and 6 edges
c 5 vertices and 14 edges d 5 vertices and 5 edges
e 5 vertices and 3 edges.
7 Determine the degree of each vertex in question 6.
8 If a graph has 5 vertices, what is the least number of edges it could have so that it is
connected?
9 Draw the following graphs:
a number of vertices is 4, deg(A) = 2, deg(B) = 3, deg(C) = 3 and deg(D) = 2
b number of vertices is 4, deg(E) = 4, deg(F) = 3, deg(G) = 2 and deg(H) = 1.
10 For this undirected graph:
a determine the number of vertices, V
b determine the number of edges, E
c determine the degree of each vertex
d determine whether or not the graph is connected
e if the graph is not connected, suggest how it may become
connected.
D
B
C
E
A
6
D
A
1
C
5
B
F
2
4
G
C
E
B
D
3
H

158 General Mathematics
11
12
13
multiple ultiple choice
The degree of the vertex, C, in this diagram is:
A 3
B 5
C 2
D 4
E 6
multiple ultiple choice
If a connected graph has 6 vertices, the least number of edges it could have is:
A 5 B 4 C 6 D 7 E 8
multiple ultiple choice
The graph at right consists of:
A 8 edges and 6 vertices
B 6 edges and 12 vertices
C 10 edges and 6 vertices
D 6 edges and 8 vertices
E 12 edges and 6 vertices
A
E D
B
A
C
D
F
B
C
E

Planar graphs
Chapter 21 Undirected graphs and networks 159
Undirected graphs that can be drawn with no crossing edges are called planar graphs
or networks. Electronic circuits are examples of planar graphs.
Sometimes it is possible to redraw undirected graphs that have crossing edges and
remove the crossovers. For example, the graph on the left may be redrawn as the graph
on the right, a planar graph.
A B
B
A
D C
To show that the above graph was planar, we simply redrew each of the vertices and
then added in the edges making sure that there were no crossovers. However, this is not
always possible.
The graph at right is not planar.
To redraw the graph, we commence with the edges from A,
and then add in the edges from B.
A B C
A B C
F E D
When we try to join C to F, we find that it must cross over an edge.
Regions
A planar graph divides the plane into a number of regions. A region,
R, is an area from which it is not possible to move unless an edge
is crossed. Regions exist within the graph as well as outside the
graph.
The graph at right has 5 vertices, 9 edges and 6 regions (that is,
5 regions from within the graph and 1 region outside the graph).
D
C
A B C
F E D
F E D
A B
1
3
5
4
E 2 D
6
C

160 General Mathematics
WORKED Example
Redraw the following networks as connected planar graphs if possible.
a A
b A
c B D
D
B
E
C
F
B C
A
C
E
G H I
D
THINK WRITE
a 1 Redraw each of the vertices. a
2 Add in the edges making sure there are
no crossovers.
D
B
A
E
C
F
3
G H I
Answer the question. The network is a connected planar graph
as there are no edges which cross over.
b Redraw each of the vertices. b
1
2
3
Add in the edges making sure there are
no crossovers.
D
Answer the question. The network is not a connected planar
graph. It is not possible to reach a
particular vertex from all other vertices.
c Redraw each of the vertices. c
1
2
3
Add in the edges making sure there are
no crossovers.
G F
Answer the question. The network is a connected planar graph
as there are no edges which cross over.
From the results obtained in worked example 3 we find that:
Figure and
type of graph
3
Number of
vertices (V)
Number of
regions (R)
B C
Number of
edges (E) V + R – E
a Planar 9 8 15 2
b Not planar 4 1 2 3
c Planar 7 7 12 2
A
B D
C
A E
G
F

Chapter 21 Undirected graphs and networks 161
In fact, if we analysed numerous planar graphs we would find, as Euler did, that in
each case the final column V + R − E would equal 2.
Leonhard Euler developed a number of important results in the theory of planar
graphs, one of which is shown below.
For any connected planar graph, the relationship between the number of vertices,
V, regions, R, and edges, E, is given by:
V + R – E = 2
This is known as Euler’s Law.
WORKED Example
4
a i Determine whether Euler’s formula holds for the given graph
and state whether the graph is a connected planar graph.
A
B
ii Compare the sum of the degrees of all the vertices and the
number of edges.
F
D
C
b
iii Establish how many odd-degree vertices there are.
Determine whether Euler’s formula holds for the given information
and state whether a connected planar graph would be produced.
E
i V = 7, R = 4, E = 8 ii V = 7, R = 5, E = 10
THINK WRITE
a i 1 Write down Euler’s formula. a i V + R − E = 2
2 Determine the number of vertices. V = 6
3 Determine the number of edges. E = 8
4 Determine the number of regions.
R = 4
5
Note: Remember there is an outside region
which must be included in the total.
Substitute the V, R, and E values into LHS = V + R − E
the LHS of Euler’s formula.
= 6 + 4 − 8
6 Evaluate. = 10 − 8
= 2
7 Compare the answer obtained with RHS = 2
the RHS of the equation.
RHS = LHS
8 Answer the question. Euler’s formula holds for the given
graph so the given graph is a
connected planar graph.
ii 1 Determine the degree of each
ii deg(A) = 2 deg(D) = 4
vertex.
deg(B) = 3 deg(E) = 2
deg(C) = 3 deg(F) = 2
2 Determine the degree of vertices sum. 2 + 3 + 3 + 4 + 2 + 2 = 16
3 Determine the number of edges of
the graph.
There are 8 edges.
4 Compare the results obtained in
The degree of all the vertices is twice
steps 2 and 3.
the number of edges.
iii Determine how many odd-degree
iii There are 2 odd-degree vertices, that
vertices there are.
is, an even number of odd-degree
vertices.
Continued over page

162 General Mathematics
THINK WRITE
b i 1 Write down Euler’s formula. b i V + R − E = 2
2 Substitute the V, R, and E values into LHS = V + R − E
the LHS of Euler’s formula.
= 7 + 4 − 8
3 Evaluate. = 11 − 8
= 3
4 Compare the answer obtained with RHS = 2
the RHS of the equation.
RHS ≠ LHS
5 Answer the question. Euler’s formula does not hold for the
given information. This combination
of values would not produce a
connected planar graph.
ii 1 Write down Euler’s formula. ii V + R − E = 2
2 Substitute the V, R, and E values into LHS = V + R − E
the LHS of Euler’s formula.
= 7 + 5 − 10
3 Evaluate. = 12 − 10
= 2
4 Compare the answer obtained with RHS = 2
the RHS of the equation.
RHS = LHS
5 Answer the question. Euler’s formula holds for the given
information. This combination of
values would produce a connected
planar graph.
As we observed in worked example 4, and as Leonhard Euler discovered:
For any connected planar graph:
• the sum of the degree of all the vertices = 2 × number of edges
• there is always an even number of odd-degree vertices.
We will work with Leonhard Euler’s results in the following exercise.
remember
remember
1. A planar graph has no crossover edges.
2. A planar graph divides the plane into a number of regions.
3. When counting regions, the region around the outside of the graph is counted
as 1.
4. For any connected planar graph:
(a) Euler’s Law states that V + R − E = 2
(b) the sum of the degree of all the vertices = 2 × number of edges
(c) there is always an even number of odd-degree vertices.

WORKED
Example
3
WORKED
Example
4a
Chapter 21 Undirected graphs and networks 163
Planar graphs
1 Redraw the following networks as connected planar graphs, if possible.
a b c d
2 a Copy and complete the table below for each of the following graphs.
i ii iii iv
v vi vii viii
i
ii
iii
iv
v
vi
vii
viii
21B
Number of
vertices (V)
Number of
regions (R)
b Copy and complete the following sentence.
For any planar graph, V + R − E = .
This formula is known as Law.
3 a Determine whether Euler’s formula holds for the
given graph and state whether the graph is a
connected planar graph.
b Compare the sum of the degrees of all the vertices
and the number of edges.
c Establish how many odd-degree vertices there are.
Number of
edges (E) V + R – E
A
D
B
C
E
F

164 General Mathematics
WORKED
Example
4b
4 Complete the tasks listed below for each of the following graphs.
a A B b A B c A
C
F
G
i Find the number of vertices.
ii Find the number of regions.
iii Write down the degree of each vertex.
iv Find the total of all these degrees.
v Write down the number of edges.
vi Compare the results obtained in parts iv and v and then copy and complete the
following sentence:
For any connected planar graph, the sum of the degrees of all the vertices =
× number of edges.
5 For each of the graphs in question 4, write down how many vertices there are whose
degree is an odd number.
6 Is the following statement true or false?
‘In any connected graph, there is always an even number of odd-degree vertices.’
7 Determine whether Euler’s formula holds for the given information and state whether
a connected planar graph would be produced.
a V = 8, R = 4, E = 10 b V = 4, R = 3, E = 5
c V = 7, R = 5, E = 10 d V = 10, R = 8, E = 15
e V = 5, R = 4, E = 6 f V = 6, R = 8, E = 4
g V = 4, R = 6, E = 8 h V = 7, R = 8, E = 13
i V = 9, R = 4, E = 11 j V = 12, R = 4, E = 14
8 Use Euler’s formula for each of the connected planar graphs below to determine the
value of the unknown.
a V = 4, R = 7, E = ? b V = 5, R = 6, E = ?
c V = 4, R = ?, E = 5 d V = 4, R = ?, E = 6
e V = ?, R = 8, E = 11 f V = ?, R = 2, E = 4
g V = 8, R = 5, E = ? h V = 5, R = ?, E = 10
i V = ?, R = 8, E = 12 j V = ?, R = 5, E = 7
9
C D
D E
d A B
e A B
f
D E
multiple ultiple choice
C
C
E F
The number of vertices a connected planar network with 6 edges and 3 regions has is:
A 1 B 5 C 4 D 6 E 3
D
B
C
D E
B
E
C
F
A
D

10
Chapter 21 Undirected graphs and networks 165
multiple ultiple choice
The number of regions a connected planar network with 8 edges and 7 vertices has is:
A 6 B 5 C 4 D 3 E 2
11 multiple ultiple choice
For the given graph ,
A V = 8, R = 5, E = 12 B V = 7, R = 5, E = 6
C V = 6, R = 6, E = 12 D V = 7, R = 6, E = 6
E V = 8, R = 6, E = 12
Schlegel diagrams
A tetrahedron (one of the 5 platonic solids) may be represented as a planar graph.
(It may help to imagine a squashed version of the tetrahedron.)
tetrahedron
Such a representation is called a Schlegel diagram.
Draw the corresponding Schlegel diagram for each of the following platonic solids.
cube octahedron dodecahedron icosahedron
Map colouring
How many different colours are needed to colour a map
so that no two adjoining regions are the same colour?
Regions that meet at a point only may be the same colour.
1 Copy and colour the ‘map’ at right using the minimum
of colours.
A
Fig 60 C
B
H
G
D
E
F

166 General Mathematics
2 The simplified map of Africa below shows borders between countries. Print it
from your Maths Quest CD and colour it using the minimum number of colours.
Africa – Political
Azimuthal Equal Area Projection
3 Suggest the minimum number of colours needed to colour any map.
N
0 500 1000 km

Eulerian paths and circuits
Chapter 21 Undirected graphs and networks 167
A path is a series of vertices connected by edges: it displays how B
to travel from one vertex to another via an edge. Just as in everyday A Fig 61
life a path is something we can walk along, we may consider a
E
path in a graph as something that can be traversed. In a path the C D
vertices are used only once and the edges are traversed (passed
over) only once; that is, edges and vertices listed in a path may not be repeated.
Some examples of paths in the graph above are: ABD, ABDE and ACDB. The
sequence ABCA is not a path since there is no edge which connects vertex B to vertex
C.
A circuit (or cycle) is a path which starts and finishes at the same vertex and no edge
is traversed (passed over) more than once.
An example of a circuit from the above graph is ABDCA.
1. A path is a series of vertices connected by edges.
2. A circuit (or cycle) is a path which starts and finishes at the same vertex and no
edge is traversed more than once.
An Eulerian path is a path which uses each edge in a graph
only once, however a vertex may be repeated.
For the graph at right, an Eulerian path could start at A,
travel to C–D–A–B–D then finish at E. Another Eulerian path
is E–D–C–A–B–D–A.
Do you remember a children’s game in which you must draw a picture
of an envelope (or a house) without lifting your pencil off the paper and
without going over any line twice?
This is an example of an Eulerian path.
A graph may have more than one Eulerian path or it may not be
possible to draw any Eulerian paths. An example of a graph for which
an Eulerian path is not possible is shown at right.
A B
Fig 62
C D
If an Eulerian path starts and finishes at the same vertex, it is 2
called an Eulerian circuit. Therefore, an Eulerian circuit is a
1
3
path which starts and finishes at the same vertex, uses all the
edges and does not go over any edge twice. An Eulerian circuit
4 5
may be drawn for the graph at right.
The Eulerian circuit is 1–2–3–5–4–3–1.
Eulerian paths and circuits have many real-life applications in areas such as planning
delivery routes and communication networks.
Consider the following example of a practical application of an Eulerian circuit.
Postal workers collect their mail from the distribution centre, deliver mail along their
particular route where each street (edge) is crossed once. At the end of their delivery
route, the postal workers return to the distribution centre. Other examples include
recycling and garbage collection, newspaper and junk mail deliveries and so on.
E

168 General Mathematics
1. An Eulerian path is a path which uses each edge in a graph only once.
2. An Eulerian circuit is an Eulerian path which starts and finishes at the same
vertex.
WORKED Example
For each of the following graphs determine whether it is possible to draw an:
i Eulerian path and
ii Eulerian circuit.
a A B E
b
Fig 66
C D
THINK WRITE
a i 1 Specify a path in which each edge is a i Begin at vertex A;
only used once.
Note: A vertex may be used more
than once.
travel to B–D–A–C–D–E.
2 Answer the question. It is possible to specify an Eulerian
path from the given graph. The
Eulerian path is
A–B–D–A–C–D–E.
ii 1 Specify a path which begins and ii Begin at vertex B; travel to
ends at the same vertex but uses
D–C–A–D–E. Cannot get back to
each edge only once.
B without going over edges already
covered.
2
3
Attempt another path.
Note: In this case any path
attempted will lead to the same
result.
5
A F E
Fig 67
B C
Begin at vertex C, travel to
D–B–A–D–E. Cannot get back to
C without going over edges already
covered.
Answer the question. It is not possible to specify an Eulerian
circuit from the given graph without
going over some of the edges twice.
b i 1 Specify a path in which each edge is b i Begin at vertex A;
only used once.
Note: A vertex may be used more
than once.
travel to B–F–D–B–C–D–E–F–A.
2 Answer the question. It is possible to specify an Eulerian
path from the given graph. This is
actually a special case of an Eulerian
path as it is also an Eulerian circuit
A–B–F–D–B–C–D–E–F–A.
D

Chapter 21 Undirected graphs and networks 169
THINK WRITE
ii 1 Specify a path which begins and ii Begin at vertex C; travel to
ends at the same vertex but uses
each edge only once.
D–E–F–A–B–F–D–B–C.
2 Answer the question.
It is possible to specify an Eulerian
Note: In this case it is possible to
circuit from the given graph. One
obtain an Eulerian circuit from any possible Eulerian circuit is
vertex.
C–D–E–F–A–B–F–D–B–C.
After studying many undirected graphs dealing with Eulerian paths and circuits, it was
found that for an Eulerian path and an Eulerian circuit to be obtained from a given network,
the following characteristics had to be satisfied:
1. An Eulerian path is possible if the number of odd-degree vertices is 0 or 2.
2. An Eulerian circuit is possible if each of the vertices has even degrees.
Let us see if these characteristics have been satisfied in worked example 5. Recall
that in worked example 5a, only an Eulerian path was satisfied. However, in worked
example 5b, both an Eulerian path and an Eulerian circuit were specified.
If we examine the degree of each vertex
in worked example 5a, we find that there Vertex Degree
are two odd number degree vertices; that is,
A and E. Therefore the characteristics for
A 3 ⇒ Odd
an Eulerian path have been satisfied. How- B 2
ever, we were unable to specify an Eulerian
circuit as each vertex did not have an even C 2
degree.
If we examine the degree of each vertex
D 4
in worked example 5b, we find that there
are 0 odd number degree vertices.
E 1 ⇒ Odd
Vertex A B C D E F
Degree 2 4 2 4 2 4
This implies that each vertex has an even degree. Therefore the characteristics for
both an Eulerian path and an Eulerian circuit have been satisfied.
remember
remember
1. A path is a series of vertices connected by edges.
2. A circuit (or cycle) is a path which starts and finishes at the same vertex and no
edge is traversed more than once.
3. An Eulerian path is a path which uses each edge in a graph only once.
4. An Eulerian circuit is an Eulerian path which starts and finishes at the same
vertex.
5. An Eulerian path is possible if the number of odd vertices is 0 or 2.
6. An Eulerian circuit is possible if each of the vertices has even degrees.

170 General Mathematics
WORKED
Example
5
Eulerian paths and circuits
1 For which of the following graphs is it possible to draw an:
a Eulerian path b Eulerian circuit?
i A B
ii A B
iii
2 Copy and complete this table for the graphs in question 1.
Vertex
A
B
C
D
E
F
Number
of odd
vertices
Eulerian
path
(Yes/No)
Eulerian
circuit
(Yes/No)
21C
D C
D C
D C
iv A B v A D
vi A C
D C
Degree of
vertex in
graph
i
Degree of
vertex in
graph
ii
Degree of
vertex in
graph
iii
Note: The shaded regions indicate that the particular vertex does not exist for the
given graph.
3 Using the results from question 2, copy and complete these rules to check if it is
possible to draw an Eulerian path or an Eulerian circuit.
a An Eulerian path is possible if the number of odd degree vertices is or .
b An Eulerian circuit is possible if the vertices all have degrees.
4 For the following networks:
i state the degree of each vertex
ii specify whether an Eulerian path is possible
E
B C
Degree of
vertex in
graph
iv
A
Degree of
vertex in
graph
v
E
F E
B
B D
Degree of
vertex in
graph
vi

Chapter 21 Undirected graphs and networks 171
iii specify whether an Eulerian circuit is possible.
a b c d
A B
G H
F
E
D
C
A
E
5 The graph at right shows a number of roads in a town. The
G
council is organising a street sweeping route so that the same road
is not used twice. The street sweeping truck is to start and finish
A D H K
at the council depot, D.
B E I
a Explain why it is not possible to do this without going down
one street twice.
C F J
b If one street were to be left out, it would be possible to complete the task without
going down the same street twice. Which street has to be left out?
6 The network at right shows a number of paths in a park.
a Is it possible to start at A and walk along each pathway once
and return to A?
b If it is, draw such a path. If not, which path needs to be
A
C
F
D
E
G
H
I
walked along twice?
B
J
7 multiple ultiple choice
For this graph, a suitable Eulerian path would be:
A B
A D–B–F–C–A
B A–C–F–C–D
C A–B–E–D–C–A
F
E
D C–F–B–D–C–A–B–E–D
C D
E C–F–B–E
Questions 8 and 9 refer to the following graphs.
i ii iii
A D
B C
8 multiple ultiple choice
The graphs which have an Eulerian path are:
A i only B i and ii only C i, ii and iii D ii and iii only E i and iii only
9 multiple ultiple choice
The graphs which have an Eulerian circuit are:
A i only B ii only C iii only D i and ii only E ii and iii only
10 multiple ultiple choice
If a graph has only even degree vertices then:
A it is possible to draw an Eulerian circuit but not an Eulerian path
B it is not possible to draw either an Eulerian circuit or an Eulerian path
C it is possible to draw an Eulerian circuit but it depends on the graph whether an
Eulerian path can be drawn
D it is possible to draw an Eulerian path but not an Eulerian circuit
E it is possible to draw an Eulerian path and an Eulerian circuit.
B
C D
A B
C D
A F E A
B D
C
B
D
C
E
B
F
A
C
E
H
D
G
WorkSHEET 21.1

172 General Mathematics
Hamiltonian paths and circuits
Eulerian paths are used when we need to find a way to travel along each edge only
once. This is useful in areas such as postal or delivery routes and garbage collections.
However, there are occasions when we are interested in travelling to each vertex only
once, but it is not important that we travel along each edge.
For example, if the vertices are five tourist areas to be visited in
a town, we may be interested in visiting each area (vertex) but not
in travelling along each road. A path that passes through each
vertex once is called a Hamiltonian path (named after Sir William
Hamilton [1805–65], a Scottish mathematician). A Hamiltonian
path must pass through each vertex once, but does not have to use each edge. Usually
all of the edges are not required to draw a Hamiltonian path.
In the graph shown (above right), one Hamiltonian path is A–B–C–D–E. Another is
A–E–D–C–B. It is possible to specify a number of Hamiltonian paths from a given graph.
A Hamiltonian path that starts and finishes at the same vertex is
called a Hamiltonian circuit, in this case, one vertex is used
twice: for starting and finishing.
In the network at right, C–D–E–A–B–C is an example of a
Hamiltonian circuit.
A B
E D
B A
1. A Hamiltonian path passes through each vertex only once. It may not use all of
the edges.
2. A Hamiltonian circuit is a Hamiltonian path which starts and finishes at the
same vertex.
WORKED Example
6
Determine which of the following have a:
i Hamiltonian path
ii Hamiltonian circuit.
a B b A B C D c A
A
C
C D
F E
E
D
THINK WRITE
B E
a i 1 Specify a Hamiltonian path for the a i Begin at vertex C; travel to D–B–A–E.
given graph.
Or begin at vertex A; travel to
Note: Each vertex must be used only
once. However, each edge does not
need to be used.
There may be more than one
Hamiltonian path.
E–D–B–C.
2 Answer the question. It is possible to specify a Hamiltonian
path from the given graph. Possible
Hamiltonian paths include:
C–D–B–A–E or A–E–D–B–C.
C
D
C
E

Chapter 21 Undirected graphs and networks 173
THINK WRITE
ii 1 Specify a Hamiltonian circuit for the ii Begin at vertex D; travel to
given graph.
Note: We must begin and end at the
same vertex and pass each of the
other vertices once only.
C–B–A–E–D.
2
Answer the question. It is possible to specify a Hamiltonian
circuit from the given graph. A
possible Hamiltonian circuit is D–C–
B–A–E–D.
b i 1 Specify a Hamiltonian path for the b i Begin at vertex A; travel to
given graph.
B–C–F–E–D.
2 Answer the question. It is possible to specify a Hamiltonian
path from the given graph. The
Hamiltonian path is
A–B–C–F–E–D.
ii 1 Specify a Hamiltonian circuit for the
given graph.
Note: To get back to vertex A, we
will have to go through vertices B
and C.
2
ii Begin at vertex A; travel to
B–C–D–E–F. It is not possible to get
back to A without passing through
B and C again.
Answer the question. It is not possible to specify a
Hamiltonian circuit as we cannot get
back to the start, that is, to vertex A,
without passing through B and C again.
c i 1 Specify a Hamiltonian path for the c i Begin at vertex A; travel to B–C.
given graph.
We cannot go any further as we are
unable to get to vertex D and E.
2 Answer the question. It is not possible to specify a
Hamiltonian path as the graph is not
connected.
ii 1 Specify a Hamiltonian circuit for the ii Begin at vertex A; travel to B–C.
given graph.
Again, it is not possible to reach
vertices D and E and then get back
to A.
2
Answer the question. It is not possible to specify a
Hamiltonian circuit as the graph is not
connected.

174 General Mathematics
The graph at right shows a delivery network with O being the
central office.
2 O 5
Gaetano must deliver items to each of the six places labelled 1, 2,
. . ., 6. The edges represent roads between the places. Plan a
delivery route for Gaetano to deliver the items to each of the six
1
3
6
places, without going to the same place twice. Gaetano must
return to the central office after the items have been delivered.
4
THINK WRITE
1 Specify a Hamiltonian path for the given
graph.
Note: Gaetano must visit each vertex only
once and start and finish at O. Therefore,
we need a Hamiltonian circuit.
Begin at vertex O; travel to 2–1–3–4–6–5–O.
Check that each vertex (except O) has Each vertex (except O) has only been used
2
3
WORKED Example
only been used once.
7
Answer the question.
Note: In this example there is more than
one possible solution.
once.
One possible solution for Gaetano’s route is
O–2–1–3–4–6–5–O. Another possible solution
is O–5–6–4–3–1–2–O.
Shortest path
The shortest path in a graph has a number of applications in business: minimising the
distance travelled, minimising the time for a journey or a series of jobs and minimising
transport costs. In shortest path problems, we usually have a starting point (for example
a depot, main office or a warehouse) and require a Hamiltonian circuit with the least
distance. One method for finding the shortest path is to move along the network choosing
the edge with the shortest distance or least cost until a Hamiltonian circuit is formed.
WORKED Example
8
A distributor supplies retail outlets labelled A, B, C, D and E from
a warehouse, W.
The distances along the roads are shown in the network at right.
Find the shortest delivery route that goes to each retail outlet and
6 km
9 km A
W 8 km
3 km
7 km
D
10 km 13 km
B
11 km
returns to the warehouse.
THINK WRITE
C
9 km
18 km
E
1 Obtain a Hamiltonian circuit starting and Begin at vertex W, travel to D–A–B–E–C–W.
finishing at W. Look at the possible routes
6 km
9 km A
WA, WD and WC. Choose the route of the
W 8 km
3 km
7 km
least distance, that is, WD. Continue from D,
D
until a Hamiltonian circuit is formed 10 km 13 km
B
11 km
2
choosing the least distance.
Highlight the selected route.
C
9 km
18 km
E
3 Answer the question. The shortest delivery route, W–D–A–B–E–C–W,
is highlighted in the above diagram.
The distance for the shortest route is 56 km,
that is, 8 + 3 + 6 + 11 + 18 + 10 = 56 km.

Chapter 21 Undirected graphs and networks 175
In other problems we may not require a Hamiltonian circuit as many shortest path
situations require simply travelling in one direction from the starting point to the
destination. A method similar to that used in worked example 8 may be used; choosing
the shortest distance edge and moving along the network until the destination is reached.
Find the shortest distance along the roads between towns A and
G in the given diagram.
THINK WRITE
1
2
3
WORKED Example
Obtain the route which gives the shortest
distance from A to G.
Note: We do not require a Hamiltonian
circuit as we are moving directly from A to
G and do not need to visit all of the towns.
Start at vertex A. Look at the possible
routes AB, AC and AD.
Choose the route of the least distance, that
is, AB.
From B, look at BG, BF and BC.
Again choose the route of the least distance,
that is, BF.
Note: The distance
from towns B to G via
F was shorter than the
more direct route of B
to G (that is, 27 km compared to 31 km).
B
9
10 km
17 km
31 km
Begin at vertex A;
travel to B–F–G.
18 km
9 km 9 km 17 km
B
Highlight the selected route.
Answer the question. The shortest delivery route from A and G,
A–B–F–G, is highlighted in the above diagram.
The distance for the shortest route is 38 km, that
is, 11 + 10 + 17 = 38 km.
There is a lot of trial and error involved in calculating the shortest distance; it is advisable
to use a pencil, eraser and scrap paper for calculations to decide the shortest path.
remember
remember
F
G
15 km
A
D
25 km
18 km
15 km
9 km 9 km 17 km
C
12 km
14 km
10 km
F
E
22 km
11 km 31 km
A
G
D
25 km
C
12 km
14 km
10 km
F
E
22 km
11 km 31 km
1. A Hamiltonian path passes through each vertex only once. It may not use all of
the edges.
2. A Hamiltonian circuit is a Hamiltonian path which starts and finishes at the
same vertex.
3. To find the shortest path in a network, choose the edge of least distance and
move along the network until the destination is reached. This usually requires a
trial and error approach.
B
G

176 General Mathematics
WORKED
Example
6
WORKED
Example
7
21D
Hamiltonian paths
and circuits
1 Determine which of the following have a:
a Hamiltonian path b Hamiltonian circuit.
i ii iii
iv v vi
2 The graph at right represents a delivery route with O being the
central office. A courier must deliver items to each of the eight
places labelled 1, 2, . . ., 8. The edges represent roads between
1
4
2
O
3
5
the places. Plan a delivery route for the courier to deliver the
items to each of the eight places without going to the same place
6
7 8
twice. The courier must return to the central office after the items have been delivered.

WORKED
Example
8
WORKED
Example
9
Chapter 21 Undirected graphs and networks 177
3 Construct a graph for which a Hamiltonian circuit is 4–3–2–1–5–4.
4 Draw a graph that does not have a Hamiltonian path.
5 Which graphs in question 1 also have an Eulerian path?
6 A distributor located at O supplies shops labelled A to F.
The distances are in kilometres. Find the shortest delivery
route from the depot at O that visits each shop and returns to
the depot.
7 A delivery firm has to collect goods at storage places E, F,
G, H and I and deliver them to the depot at D. Find the
shortest route from the depot visiting each storage place and
returning to the depot.
20 km
8 Find the length of the shortest Hamiltonian circuit in each of the following.
a 4
b 6
c 11 d 7
5
3 3 7
10 5 7 7
5
4 4 5 5
4
4
5
4 4
8
12 10 10
7
6 6
9
4
6
10
4 3
12
8
(Hint: You may find it easiest to start in the top left-hand corner of each network.)
9 The table shows the distance, in km, between storage points A, B, C, E, F and G and
a depot D.
A B C E F G
D 8 12 15 16
A 15 12
B 10 22
C 29 14
E 15
F 19
Note: The shaded region indicates that there is no road connecting
the two towns.
A
D E
a
b
Transfer the information from the above table onto a copy of
the graph at right.
Find the shortest path from the depot, visiting each storage
point and finishing at the depot.
B
C F
G
10 Find the shortest distance along the roads, between the towns S and F in each of the
folowing diagrams. (Note that distances are in kilometres.)
a 8
A
B
5 6 9
3 6
S
10
6
C
3
D F
7 7
8
E
G
8
b
4
9
7
5 13
6
18 7
4
7
10
c
21
A
S
C
B
10
E G F
10
D
6
S
6
C
10
A
25
12
12
6
E
B
D
12
19
17 F
10
11
G
A
6
B
D
11 km
O
C
D 10
10
3
17 5 5
5
E
11 7
22
16 km
16
F
8 km
15 km
8 km
E
7 km
F
10 km
I
6 km
H
12 km
G
9 km

WorkSHEET 21.2
178 General Mathematics
11 For the network shown in the diagram at right, find the
shortest distance between:
a A and H b E and G.
12 A tour of six country towns is to start and finish at
Bendigo. Find the shortest distance tour.
13 The distance, in km, between five towns is shown in the table below.
B C D E
A 300 353 417 280
B 219 453 345
C 291 402
D 258
Note: The shaded region indicates that there is no road connecting the two towns.
a Transfer the information from the above table onto a copy of the A
graph at right.
B
b A tour is planned to start and finish at A and visit all of the towns. E
c
Find the shortest distance for the tour.
If the tour were to start and finish at C, would the shortest distance
be the same?
D
C
d A new tour, starting and finishing at A, is planned to visit all the towns and a
further town, F. The distances from B to F, C to F and E to F are 476, 429 and
319 km respectively. Find the shortest distance tour now possible.
14 multiple ultiple choice
A Hamiltonian circuit for the graph at right is:
A A–D–E–C–F–G–A B A–D–B–C–E–C–F–G–A
C A–D–E–C–B–F–G D A–D–E–C–B–F–G–A
E A–G–F–B–D–C–E–A
15 multiple ultiple choice
The length of the shortest Hamiltonian circuit is:
A 53 km B 47 km C 56 km
D 40 km E 51 km
10
A
E
14
6 10
C
13
13
11
12
22
B
8
G
5
9 H
9 4 D
F
Mildura
225 km
103 km
Ouyen 140 km
Swan Hill
60 km
212 km
Kerang
115 km
Horsham
219 km Bendigo
13 km
E
G
247 km
234 km
A D
B
F C
2 km
D
15 km
A 3 km B
5 km
17 km
F
9 km
12 km
14 km
E
12 km
C

Trees
Chapter 21 Undirected graphs and networks 179
A tree is a connected graph without any circuits, loops or multiple edges.
3
B
Examples of trees include: 1 2 4 and C A D
5
E
Since a tree cannot have any circuits or loops, it contains only one region.
Examples of graphs which are not trees include:
and
A B
B A
2
1
C D D C
4 5
This is not a tree This is not a tree This is not a tree
because it is because the graph because the graph
a circuit. contains a loop. contains a multiple edge.
Trees are used in transportation and communication networks, in computer programming,
in planning projects to represent independent activities or structures, and in
road and railway planning.
WORKED Example
10
Which of the following are trees?
a 1 2
b 1 2
c
1 2
4 3
3 4
3 4
5 6
THINK WRITE
a Determine whether the graph is connected. a This graph does not represent a tree since
it is not connected.
b 1 Determine whether the graph is
connected.
b The graph is connected.
2 Determine whether the graph contains This graph does not represent a tree since
any circuits, loops or multiple edges. it contains a circuit, that is, 1–2–4–1.
c 1 Determine whether the graph is connected. c The graph is connected.
2 Determine whether the graph contains This graph does represent a tree as it
any circuits, loops or multiple edges. meets each of the criteria, that is, this is a
Count the number of regions.
connected graph without any circuits,
loops or multiple edges. There is only
1 region.
In order for any network to be defined as a tree, it must satisfy the following
criteria:
1. the graph must be connected
2. the graph must not contain any circuits, loops or multiple edges
3. the graph must contain only one region.
3

180 General Mathematics
WORKED Example
11
a Remove the edges from this graph to produce a tree.
b Comment on the relationship between the number of edges and the
number of vertices.
THINK WRITE
a 1 Look at the given graph and identify a There are 4 circuits: A–B–C–A,
any circuits.
B–D–E–B, B–E–F–B and B–D–E–F–B.
2 Remove one of the edges from circuit
A–B–C–A, say BC.
A
B
D
E
3 Remove the edge BE from circuit
A
B–D–E–F–B
4 Remove the edge BF from circuit
A
B–D–E–F–B.
b 1 Count the number of vertices, V. b V = 6
2 Count the number of edges, E. E = 5
3 Answer the question. The difference between the vertices and
edges in a tree is 1; that is, V − E = 1.
C
C
C
The tree obtained in worked example 11 is one possible spanning tree for the graph. A
spanning tree is a tree that includes all the vertices in the graph. Find other spanning
trees in worked example 11 by removing different edges.
Often it is necessary to find a minimal spanning tree; that is, a spanning tree with the
minimum length (or cost, or time).
To find a minimal spanning tree:
1. select the edge with the minimum value. If there is more than one such edge,
choose any one of them.
2. select the next smallest edge, provided it does not create a cycle.
3. repeat step 2 until all the vertices have been included.
F
D
B E
F
D
B E
F
A
C
D
B E
F

WORKED Example
Chapter 21 Undirected graphs and networks 181
a Find the minimal spanning tree of the network shown at right.
b Comment on the relationship between the number of edges and
the number of vertices.
THINK WRITE
A
B
8
6 10
C
6
4
D
3 E 5
12
14
7
F
a 1 Select the edge with the smallest value a The smallest edge is CE.
and highlight it.
B
2
3
4
5
Select the next smallest edge and
highlight it.
Select the next smallest edge and
highlight it.
A C D F
3 E
The next smallest edge is CA.
A
4
C
3 E
D F
The next smallest edge is ED.
A
4
C
3 E
D
5
F
Continue this process until vertices B
and F are connected and highlighted.
Note: BC and DF respectively are the
next smallest edges.
A
4
B
6
C
3 E
D
5
7
F
Answer the question. The minimal spanning tree is shown
above. Its value is 4 + 6 + 3 + 5 + 7 = 25.
b Count the number of vertices, V. b V = 6
1
2
3
12
Count the number of edges, E. E = 5
Answer the question. The difference between the vertices and
edges in a tree is 1; that is, V − E = 1.
remember
remember
1. A tree is a connected graph without any circuits, loops or multiple edges which
contains only one region.
2. A spanning tree is a tree that includes all the vertices in the graph.
3. A minimal spanning tree is a spanning tree with the minimum length (or cost,
or time).
4. To find a minimal spanning tree:
(a) select the edge with the minimum value. If there is more than one such
edge, choose any one of them.
(b) select the next smallest edge, provided it does not create a cycle.
(c) repeat step b until all the vertices have been included.
5. The vertices and edges in a tree are related by the equation V − E = 1.
B
B

182 General Mathematics
WORKED
Example
10
WORKED
Example
11
WORKED
Example
12
21E
Trees
1 Which of the following are trees?
a b c d
e f g h
For those networks which are not trees, give reasons why they are not.
2 Change those graphs in question 1 that are not trees into trees by removing or adding
edges.
3 a i Draw a tree with 5 vertices. ii How many edges are there?
b i Draw a tree with 11 vertices. ii How many edges are there?
c Copy and complete: If a tree has V vertices, the number of edges is given by
E = .
4 Use the result from question 3 to find how many vertices there are in a tree with:
a 8 edges b 9 edges c 16 edges d 17 edges e 20 edges
5 Construct spanning trees for the following graphs:
a b c d
6 Find the value of the minimal spanning tree for each of the following graphs.
a
1
2
6
8
3
5
4
b
3
6
5
5
3
2
4
c
5
2
7
6
4
8
4
7
4
d 5
e 9
f 6
5
4
4
3
3
5
5
5
6
5
5
4
2
4
7
6
4
5
5
5
3
7 4
6 3
6
4 7
5 4
5 6
7
3
4
5
3
5
7 A communication network is to be developed linking six
towns. The distances (in km) between the towns are shown
in the graph at right. Calculate the minimal spanning tree so
that the length of cabling used to connect the towns is a
minimum.
58
62
97
52
47
76
80
143
84
165
143

8
9
Chapter 21 Undirected graphs and networks 183
multiple ultiple choice
The following graphs which represent trees are:
i ii iii iv
A i and iv only B i and ii only C ii and iii only
D iii and iv only E all of them
multiple ultiple choice
The network at right, when changed to a tree, will resemble which
of the following?
A C
B C
C
A
B
E
A
B
E
D
F
D
F
D C
E
A
E
B
D
F
A
B
C
D
F
E
A
D
F
A
F
C
B
C
D
E
B
E

184 General Mathematics
summary
Vertices and edges
• An undirected graph or network consists of vertices and edges.
• The degree of a vertex is the number of edges leading
to or from that vertex. A loop counts as 2 edges.
• In a connected graph it is possible to reach each vertex
from any other vertex. A connected graph must not
have any isolated vertices.
Planar graphs
• A planar graph has no crossover edges.
• A planar graph divides the plane into a
Multiple
edges
number of regions.
• When counting regions, the region around A planar graph
the outside of the graph is counted as 1.
Not a planar graph
• For any connected planar graph: V = 5
Euler’s Law states that: V + R − E = 2. E = 8
The sum of the degree of all the vertices = R = 5
2 × number of edges. V + R − E
There is always an even number of odd-degree = 5 + 5 − 8
vertices.
Eulerian paths and circuits
• A path is a series of vertices connected by edges.
= 2
• A circuit (or cycle) is a path which starts and finishes at the same vertex and no edge
is traversed more than once.
• An Eulerian path is a path which uses each edge in a graph only once, however a
vertex may be repeated.
• An Eulerian circuit is an Eulerian path which starts and finishes at the same vertex.
• An Eulerian path is possible if the number of odd vertices is 0 or 2.
• An Eulerian circuit is possible if each of the vertices has even degrees.
Hamiltonian paths and circuits
• A Hamiltonian path passes through each vertex only once. It is not necessary to use
all of the edges.
• A Hamiltonian circuit is a Hamiltonian path that starts and finishes at the same vertex.
• To find the shortest path in a network, choose the edge of least distance and move
along the network until the destination is reached. This usually requires a trial and
error approach.
Trees
• A tree is a connected graph without any circuits, loops or multiple edges and
contains only one region.
• A spanning tree is a tree that includes all the vertices in the graph.
• A minimal spanning tree is a spanning tree with the minimum length (or cost, or time).
• To find a minimal spanning tree:
1. select the edge with the minimum value. If there is more than one such edge,
choose any one of them.
2. select the next smallest edge, provided it does not create a cycle.
3. repeat step 2 until all the vertices have been included.
• The vertices and edges in a tree are related by the equation V − E = 1.
Loop
Edge
Vertex

CHAPTER review
Multiple choice
1 Which of these graphs is not a connected graph?
A B C
D E
Questions 2 and 3 refer to the diagram at right.
2 The vertices with degree 4 are:
A A and B B B only C B and D
D all of them E none of them
Chapter 21 Undirected graphs and networks 185
3 The number of edges in total is:
A 5 B 4 C 7 D 9 E 8
4 For the graph at right:
A V = 4, E = 6, R = 4 B V = 6, E = 4, R = 3
C V = 4, E = 6, R = 3 D V = 4, E = 5, R = 4
E V = 6, E = 4, R = 4
5 The number of regions that a connected planar network with 12 edges and
8 vertices has is:
A 7 B 6 C 5 D 4 E 3
6 The description that does not satisfy Euler’s formula for a planar network is:
A V = 6, E = 7, R = 3 B V = 9, E = 15, R = 8 C V = 12, E = 21, R = 12
D V = 9, E = 12, R = 5 E V = 5, E = 11, R = 8
Questions 7 and 8 refer to the following graphs.
i A B
ii A B
iii A B
C D
iv v
B
A A B
C
D
E F
G
C
D E
C
D E
B C
A
E C
D
D
21A
21A
21A
21B
21B
21B

21C
21C
21C
21D
21D
21C,D,E
21E
21E
186 General Mathematics
7 The graph that has an Eulerian path is:
A i only B ii only C iii only D iv only E v only
8 The graph that has an Eulerian circuit is:
A i only B ii only C iii only D iv only E v only
9 If a graph has 2 vertices of odd degree and all the other vertices of even degree, then:
A it is possible to draw an Eulerian path and an Eulerian circuit
B it is possible to draw an Eulerian path but not an Eulerian circuit
C it is possible to draw an Eulerian circuit but not an Eulerian path
D it is not possible to draw either an Eulerian circuit or an Eulerian path
E it is possible to draw an Eulerian circuit but it depends on the graph whether an Eulerian
path can be drawn.
10 A Hamiltonian circuit for the graph at right is:
A 1–7–6–5–4–3–2
B 6–7–5–6–1–2–3–4
C 1–2–3–4–5–7–6–1
D 6–7–5–4–3–2–6–1–6
E 1–2–3–4–6–5–7–1
11 The length of the shortest Hamiltonian circuit is:
A 35
B 32
C 33
D 17
E 34
12 A distributor supplies four shops from a warehouse. The distances of the shops from the
warehouse range from 10 km to 20 km. The shortest delivery route from the warehouse to
13
all four shops and back to the warehouse could be found by using:
A an Eulerian path B an Eulerian circuit C a Hamiltonian path
D a Hamiltonian circuit E a minimal spanning tree.
i ii iii iv
The graphs that are trees are:
A all of them B i only C i and ii only
D i and iv only E iii only
14 The minimal spanning tree of the graph at right has a length of:
A 24
B 26
C 19
D 15
E 25
A
1
7
2
6
5
3
6
A 8 B
4
E
9
6
D
5
7
7
4
C
B
8
4
6
C
7
8
11
D
3
E
14 10
F
5

Chapter 21 Undirected graphs and networks 187
15 When converted to a spanning tree, the network at right will resemble
which of the following?
A A B B
E F
E F
C A B D A B E
C
C
E F
Short answer
D
D
A B
1 For the graph at right:
a write down the number of vertices
b write down the number of edges
c write down the degree of each vertex
d establish if the graph is connected
e if it is not connected, suggest how it may become connected.
2 Draw a connected graph with 10 vertices, 18 edges and 2 loops.
3 Re-draw the graph at right to show clearly that it is a planar graph.
4 a Using Euler’s Law, determine whether the following would produce a connected planar
graph.
i V = 3, R = 2, E = 3 ii V = 7, R = 4, E = 8
iii V = 10, R = 12, E = 20 iv V = 6, R = 5, E = 9
b Use Euler’s Law for each of the connected planar graphs below to determine the value of
the unknown.
i V = 10, R = 9, E = ? ii V = 8, R = 7, E = ?
iii V = 4, R = ?, E = 3 iv V = ?, R = 4, E = 5
5 For the graph at right:
a how many vertices and edges are there in the graph?
b write down the degree of each vertex
c explain how your answer to b means that the graph does not
have an Eulerian circuit
d write down an Eulerian path for the graph.
C
C
E F
D
D
C
C
A B
E F
A
D
A B
E F
B
E
D
D
21A
G 21A
C
F
H
I
A D
A
B
E
D
C
F
B C
E
F
21A
21B
21B
21C

21C
21D
21D
21E
21E
188 General Mathematics
6 Trish delivers the local paper after school each Tuesday afternoon.
On most days she collects the papers from the printers and
M A
H
D
I L
distributes them along the route illustrated at right.
a i If Trish is able to start from any point, is she able to deliver
the papers going down each of the streets only once?
ii If she is, write down one possible route she may take.
B
C
E
F
J
K
b
iii Is this the only possible route? If not, write down an alternative route.
i If the papers are dropped off at Trish’s house, (vertex L), is she able to complete her
round and return home going down each of the streets only once?
ii If so, which type of circuit has been completed?
iii Write down two possible circuits that Trish could have completed.
7 a Write down two Hamiltonian circuits starting at vertex 1 for the
network at right.
b Calculate the least value of the Hamiltonian circuit for the graph.
c Give one real-life situation for which the least value Hamiltonian
circuit found above would be appropriate.
d Is a Hamiltonian circuit possible for the graph in question 5?
8 The distance, in km, between five towns is shown in the table below.
B C D E
A 400 462 489 310
B 290 495 420
C 320 470
D 300
Note: The shaded region indicates that there is no road connecting the two towns.
a Transfer the information from the above table onto a copy of the
graph at right.
A
b A tour is planned to start and finish at A and visit all of the towns.
Find the shortest distance for the tour.
E
B
c
d
If the tour were to start and finish at C, would the shortest distance
be the same?
A new tour, starting and finishing at A, is planned to visit all the
D
C
towns and a further town, F. The distances from B to F, C to F and E to F are 480, 429
and 339 km respectively. Find the shortest distance tour now possible.
9 a Find a spanning tree for the graph at right.
b How many edges are there in a tree with:
i 6 vertices?
ii 11 vertices?
iii 14 vertices?
c Draw an example of each of the trees in part b.
10 Find the shortest distance from S to F in the following network.
8
1
2
6
10
9
B
5
15
3
14
7
13
8
4
6
10
A D
E
C
10
A
S
6
12
14
16
12
B
8
14
D
E
18
12
23
F
G 15
C 16
F

Chapter 21 Undirected graphs and networks 189
Analysis
The map shows six towns, labelled A to F, on an island. The lines
F
represent roads linking the towns, with distances given in
kilometres. The map may be treated as a planar network with the
towns as vertices and the roads as edges.
a What is a planar network?
A
12
18
22
21
14
E
21
D
b
c
Explain what the statement ‘the degree of F = 3’ means.
Write down the degree of each vertex.
B
20 C
19
d Calculate S, the sum of the degrees of all the vertices.
e How many edges are there?
f Write down a formula linking S and E for this network (E = number of edges).
All the roads on the island are to be re-surfaced, with the work starting at F.
g Give a route to be used by the workers so that each road is re-surfaced and no road is
travelled twice.
h Is the route above an example of:
i an Eulerian circuit?
ii an Eulerian path?
iii a Hamiltonian circuit?
iv a Hamiltonian path?
i Calculate the shortest distance from A to D.
A communication network is to be established on the island.
j Explain why a minimal spanning tree for the network would be used in planning the
communication network.
k Calculate the minimal spanning tree for the network.
A supermarket chain has stores at each of the towns. Each store is to be visited by the regional
manager, starting and finishing at the main office in B.
l Outline the shortest route (that is, the least distance to travel).
m Is the route above an example of:
i an Eulerian circuit?
ii an Eulerian path?
iii a Hamiltonian circuit?
iv a Hamiltonian path?
test
test
yourself
ourself
CHAPTER
21