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C H A P T E R
23
MODULE 5
E
Undirected graphs
PL
How do we represent a graph by a diagram and by a matrix representation?
How do we define each of the following:
r graph
r subgraph
r vertex
r edge (node)
r loop
r isolated vertex
r bipartite graph
r degenerate graph
r connected graph
r Euler path
r Euler circuit
r circuit
r tree
r Hamilton path
r Hamilton circuit
r adjacency matrix
r spanning tree
r complete graph
r planar graph
r degree of a vertex
r simple graph
M
How do we apply Euler’s formula to planar graphs?
How do we determine the shortest path between two given vertices on a graph?
How do we find the minimum spanning tree of a graph?
How do we apply Prim’s algorithm to find the minimum spanning trees of a
connected graph?
SA
How do we apply the Hungarian algorithm?
23.1
Introduction and definitions
A hockey team belongs to a competition in which there are six teams: A, B, C, D, E and F.
A few weeks into the season, some of the teams have played each other:
A has played B, F and E
F
B has played C, F and A
B
C has played B, D and E
A
D has played C and F
C
E has played A and C
E
F has played A, B and D
D
Figure 23.1
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Chapter 23 – Undirected graphs
615
A

A 0

B 1

C
0
D
0

E 1
F 1
B
1
0
1
0
0
1
C
0
1
0
1
1
0
D
0
0
1
0
0
1
E
1
0
1
0
0
0
PL
A B C D E F
A 0 1 0 0 1 1
B 1 0 1 0 0 1
C 0 1 0 1 1 0
D 0 0 1 0 0 1
E 1 0 1 0 0 0
F 1 1 0 1 0 0
E
This situation can be represented graphically. The teams can be represented by points and two
such points are joined by a line whenever the teams they represent have played each other. A
figure such as Figure 23.1 is called a graph.
The points A, B, C, D, E and F are called the vertices or nodes of the graph and the lines
connecting the vertices are called edges.
A graph consists of a set of elements called vertices and a set of elements called edges.
Each edge joins two vertices.
The vertices A and B of a graph are adjacent vertices if they are joined by an edge.
The graph can be represented by a table or a matrix. A ‘1’ is used to denote that there is
one edge connecting the two vertices and a ‘0’ indicates that there is no edge. The matrix
shown below is called an adjacency matrix.
Table for figure 23.1
F

1

1

0

1


0
0
Matrix for figure 23.1
M
The adjacency matrix A of the graph is an n × n matrix in which for example the entry in
row C and column F is the number of edges joining vertices C and F.
SA
Another graph, Figure 23.2, is shown opposite. Note that
three edges leave from A. The graph is said to have
multiple edges as there are two edges joining A and D.
C has one edge, which links C to itself. This edge is called
a loop. The table and matrix appear as shown below.
A
C
D
B
Figure 23.2
Note:
A loop is recorded as one edge in an adjacency matrix
A B C D
A 0 1 0 2
B 1 0 0 1
C 0 0 1 0
D 2 1 0 0
A

A 0
B
1

C 0
D 2
B C D

1 0 2
0 0 1


0 1 0
1 0 0
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Essential Further Mathematics – Module 5 Networks and decision mathematics
Example 1
Representing a graph by a table or matrix
Construct the table and matrix corresponding to the graph
shown, which represents three houses, A, B and C,
connected to three utility outlets, gas (G), water (W)
and electricity (E).
A
B
C
G
W
E
Solution
A

A 0

B 0

C 
0
G 
1

W 1
E 1
B C G W
0 0 1 1
0 0 1 1
0 0 1 1
1 1 0 0
1 1 0 0
1 1 0 0
PL
A B C G W E
A 0 0 0 1 1 1
B 0 0 0 1 1 1
C 0 0 0 1 1 1
G 1 1 1 0 0 0
W 1 1 1 0 0 0
E 1 1 1 0 0 0
E
Figure 23.3
E

1

1

1

0


0
0
Figure 23.3 is called a bipartite graph as the set of vertices is partitioned into two disjoint sets
(A, B, C) and (gas, water, electricity) and each edge has a vertex in each set.
M
A bipartite graph is a graph whose set of vertices can be split into two subsets X and Y in
such a way that each edge of the graph joins a vertex in X and a vertex in Y.
At each non-isolated vertex, A, in a graph there will be some edges joined to A (incident with
vertex A).
The degree of vertex A, written as deg(A) is the number of edges incident with it.
SA
A
For Figure 23.4:
deg A = 3
deg B = 2
deg C = 2
Note:deg D = 5
D
C
B
Figure 23.4
Loops add two to the degree of avertex.
A simple graph is a graph with no loops or multiple edges.
Figure 23.5 is a simple graph. For a simple graph,
the sum of the degrees of the graph is equal to
twice the number of edges of the graph.
∴ the sum of the degrees of a simple graph is even.
E
D
A
B
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Chapter 23 – Undirected graphs
617
For Figure 23.5:
Sum = deg(A) + deg(B) + deg(C) + deg(D) + deg(E)
= 3 + 2 + 2 + 3 + 2 = 12
For a simple graph, the entries for the corresponding table are
either 1s or 0s.
Note that the sum of the 1s = 12 = 2 × the number of edges.
E
A B C D E
0 1 0 1 1
1 0 1 0 0
0 1 0 1 0
1 0 1 0 1
1 0 0 1 0
B
PL
If E is an isolated vertex (i.e. there are no edges
incident to E) then deg(E) = 0. See Figure 23.6.
A
B
C
D
E
A
C
E
D
Figure 23.6
B
A graph is said to be degenerate if all its vertices
are isolated. See Figure 23.7.
A
C
M
Figure 23.7
SA
Many applications of graphs involve ‘getting from one vertex to another’. In order to discuss
B
this, the idea of a path is introduced.
C
A path in the graph in Figure 23.8 from A to F could be
A
D
AB, BG, GE, EC, CD, DE, EG, GF (shown in red). Not all
G
the edges or vertices in a path are required to be different.
E
F
A path can be thought of as a sequence of edges.
Figure 23.8
A graph is said to be connected if there is a path between each pair of vertices.
The graph in Figure 23.9 is connected. The graph in Figure 23.6 is not connected.
A circuit is a sequence of edges linking successive vertices
that starts and finishes at the same vertex.
In Figure 23.9, one circuit is A B C D F A. Another
circuit is E F D E.
B
C
A
D
F
E
Figure 23.9
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Essential Further Mathematics – Module 5 Networks and decision mathematics
A subgraph of a graph consists of selected edges and vertices of the graph with the same
links as the original graph and with the selected vertices including all the endpoints of the
selected edges.
Figures 23.10a and 23.10b show two subgraphs of Figure 23.9.
B
B
C
A
B
C
A
C
A
D
D
F
E
E
F
D
F
E
E
Figure 23.10b
Figure 23.10a
Figure 23.10c
PL
The graph shown in red in Figure 23.10c is not a subgraph as it has an edge that does not exist
in the original graph.
Exercise 23A
Town B
1 This section of a road map can be considered
as a graph, with towns as vertices and the
roads connecting the towns as edges.
Town C
Town A
Town D
M
a Give the degree of:
i Town A
ii Town B
iii Town H
b Construct the table (matrix) for this graph.
c Is this graph simple? Why?
Town H
2 For each of the following graphs give the associated table (matrix).
a
A
D b
SA
B
A
B
C
c A
B
C
D
d
A
D
C
e A
D
B
f
A
g
B
E
C
C
D
A
B
B
h
C
D
F
A
C
3 Which of the graphs in Question 2 are:
a simple?
b connected?
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Chapter 23 – Undirected graphs
619
4 For each of the following, draw the graph for the given table (matrix).
A
B
C
D
A
0
1
1
0
B
1
0
1
1
d 1
0


0
0
0
0
1
1
0
1
1
0
C
1
1
0
0

0
1


0
0
D
0
1
0
0
b
A
B
C
e

0

1
1
A
0
0
0
B
0
0
0
1
0
1

1

1
0
C
0
0
0
c
A
B
C
D
A
0
1
0
1
B
1
0
1
0
f 0
1


2
1
1
0
1
1
2
1
0
0
C
0
1
0
1

1
1


0
0
PL
5 a Draw three connected subgraphs, with four
B
vertices, of the graph shown. (Note that there are
many such subgraphs.)
A
b What is the degree of vertex A?
c Give the sum of the degrees of the vertices of the
graph.
F
d Give a circuit from A passing through 4 vertices (including A).
C
D
E
M
Planar graphs and Euler’s formula
Leonard Euler (pronounced ‘oiler’) was one of the most prolific mathematicians of all time.
He contributed to mathematics in an amazing array of topics. His proof of the formula named
after him is considered to be the beginning of the branch of mathematics called topology. The
application of Euler’s formula is developed in this section.
Electrical circuits can be represented by a graph, as in
C
B
Figure 23.11.
SA
23.2
D
1
0
1
0
E
a
D
A
F
E
Figure 23.11
For certain electrical circuits it is advantageous to not have connections AD and CE crossing.
C
B
Figure 23.11 can be redrawn as shown in Figure 23.12 so
that these edges do not cross.
D
A
F
E
Figure 23.12
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Essential Further Mathematics – Module 5 Networks and decision mathematics
A graph that can be drawn in such a way that no two edges meet (or have common points),
except at the vertices where they are both incident, is called a planar graph.
E
Not all graphs are planar. Figure 23.13 cannot be redrawn
so that the edges have no intersection points except at the
vertices. It is true that all simple graphs with four or fewer
vertices are planar.
PL
Figure 23.13
C
B
f4
f1
A
f2
D
f3
F
E
Figure 23.14
M
Consider a connected planar graph ABCDEF shown in
Figure 23.14. This graph has 6 vertices and 8 edges.
Connected planar graphs also have faces or regions.
Figure 23.14 has four faces: f 1 , f 2 , f 3 and f4 , where
f4 is an infinite face.
There is a relationship between the number of vertices,
edges and faces in a connected planar graph.
Let v denote the number of vertices.
Let e denote the number of edges.
Let f denote the number of faces.
Note that in Figure 23.14:
SA
v −e+ f =6−8+4=2
This result holds for any connected planar graph and is known as Euler’s formula.
Euler’s formula states:
v −e+ f =2
Example 2
Verifying Euler’s formula
Verify Euler’s formula for the graph shown.
C
B
E
f4
f3
f1
A
Solution
f2
D
1 The vertices are A, B, C, D, E. There are 5 vertices so v = 5. v = 5
2 There are 4 faces as shown on the diagram.
f=4
Don’t forget the infinite face f1 : So f = 4.
3 The edges are AB, BC, CD, DA, DB, AE and CA.
There are 7 of them, so e = 7.
e=7
v−e+ f =5−7+4=2
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Chapter 23 – Undirected graphs
621
Example 3
Verifying Euler’s formula
A tetrahedron may be represented graphically by a connected planar graph.
Verify Euler’s formula for the graph shown.
Solution
Example 4
E
v = 4, f = 4, e = 6
v−e+ f =4−6+4=2
Using Euler’s formula
PL
A connected planar graph has 6 vertices and 8 edges. How many faces does the graph have?
Draw a connected planar graph with 6 vertices and 8 edges.
Solution
Euler's formula: v − e + f = 2
v = 6 and e = 9
6−9+ f =2
−3 + f = 2
f =5
B
C
f1
f2
A
D
f3
E
f4
f5
M
F
Exercise 23B
SA
1 For each of the following graphs:
i state the values of v, e and f
a
ii verify Euler’s formula.
c
b
d
2 Show that the following graphs are planar by redrawing them in a suitable form:
a A
B
C
B
b
A
F
E
c A
C
D
D
B
C
E
D
F
E
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Essential Further Mathematics – Module 5 Networks and decision mathematics
3 For a planar connected graph, find:
a f, if v = 8 and e = 10
c f, if v = 5 and e = 14
b v, if e = 14 and f = 4
d e, if v = 10 and f = 11
23.3
E
4 a Represent this cube as a planar graph.
b Verify Euler’s formula for this graph.
Complete graphs
K3
PL
A complete graph is a graph with edges connecting all pairs of vertices. The complete
graph with n vertices is denoted by K n .
K4
K5
Note: K 3 and K 4 are planar while K 5 is not.
n(n − 1)
edges.
2
M
The complete graph with n vertices, K n , has
For example:
4(4 − 1)
= 6 edges
2
5(5 − 1)
= 10 edges
K 5 has
2
K 4 has
SA
A complete graph could be used to represent a
round robin competition (a competition in
which each side plays every other side once). A
For example, for five teams, A, B, C, D and E
are shown. Each edge represents a match that
D
has been played. The adjacency matrix has
1’s in all positions except the main diagonal.
This graph could be drawn for six people
at a party. Each edge indicates that a
conversation took place. The adjacency
matrix has 1s in all positions except the
main diagonal.
B
B
C
E

C
A
D
F
E
A
A 0

B 1

C
1

D 1
E 1
0

1

1

1


1
1
B
1
0
1
1
1
C
1
1
0
1
1
1
0
1
1
1
1
1
1
0
1
1
1
D
1
1
1
0
1
1
1
1
0
1
1
E
1

1

1


1
0
1
1
1
1
0
1
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
1

1

1

1


1
0
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Chapter 23 – Undirected graphs
623
Exercise 23C
1 A competition consists of six teams. In the competition each team plays every other team.
a How many matches are there?
b Represent the competition with the complete graph K 6 .
c Represent the competition with an adjacency matrix.
b Draw K 7 .
E
2 a How many edges does K 7 have?
3 How many handshakes could take place among 8 people. Represent this with a graph.
Euler paths
PL
Euler and Hamilton paths
In a paper published in 1736, Leonard Euler solved the problem that had intrigued the citizens
of Königsberg, which lay near the mouth of the Pregel River. The river divided the city into
four parts, as shown in Figure 23.15. There were seven bridges.
A
B
D
Pregel River
M
C
Figure 23.15
The problem was posed as follows:
‘Could we walk over each of the seven bridges once only and return to our starting point?’
Euler represented this situation as shown in Figure 23.16.
A
Euler showed that this graph cannot be traversed
B
completely in a single cyclical trail. That is, no matter
D
which vertex is chosen as the starting point it is
impossible to cover the graph and come back to the
chosen starting vertex while using each
C
edge only once.
Figure 23.16
SA
23.4
We recall that for a graph, a path is a sequence of edges.
In Figure 23.17, one path from A to D is e1 , e3 , e4 (red).
Another path from A to D is e5 (green).
B
e1
A
e3
C
e4
e2
e5
Figure 23.17
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Essential Further Mathematics – Module 5 Networks and decision mathematics
Recalling the definition of circuit in section 23.1: a
circuit is a path that begins and finishes at a particular
vertex. In Figure 23.18, one circuit is e1 , e2 , e3 , e4 ,
e5 , e6 (red). This circuit begins and finishes at A.
e2
B
e1
C
A
e3
D
e6
e5
e4
E
F
Figure 23.18
E
A path that includes every edge just once is called an Euler path.
An Euler circuit is an Euler path that starts and finishes at the same vertex.
Identifying Euler paths and circuits
Example 5
PL
A useful way of identifying Euler circuits is to look for a connected graph where all
vertices have an even degree.
The converse result also holds: if a graph has an Euler circuit, then it is connected and each
vertex has an even degree. Using this result, it can be seen that the graph for the Königsberg
bridge problem does not contain an Euler circuit as vertex B has an odd degree.
A connected graph has an Euler path starting at vertex A and finishing at vertex B if A
and B are the only odd vertices of the graph.
D
Identifying an Euler circuit
a Explain why the graph shown opposite has an Euler circuit.
b List an Euler circuit for this graph.
M
C
E
Solution
B
a The graph has only even vertices.
b An Euler circuit is C → D → E → C → A → B → C .
SA
Example 6
A
Identifying an Euler path
a Explain why the graph opposite has an Euler path
between B and C.
b List one possible Euler path (there are several)
between B and C.
Solution
A
E
B
D
C
a B and C are odd vertices. The remaining vertices are even.
b B → A → E → D → B → C → D → C.
Hamilton paths
A Hamilton path is a path through a graph that passes through each vertex exactly once.
A Hamilton circuit is a Hamilton path which starts and finishes at the same vertex.
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Chapter 23 – Undirected graphs
625
It is easy to remember the difference between Hamilton paths (circuits) and Euler paths
(circuits). Hamilton graphs are defined in terms of vertices and Euler graphs are defined
in terms of edges.
Unfortunately, unlike the condition for an Euler circuit, there is no nice condition to identify
when a graph is a Hamilton circuit. It is just a matter of trial and error.
Example 7
B
C
A
E
Identifying a Hamilton circuit
List a Hamilton circuit for the graph shown.
E
D
Solution
H
F
G
PL
A Hamilton circuit is C → B → A → E → F → G → H → D → C .
Not every graph that has a Hamilton circuit has an Euler circuit,
and also not every graph that has an Euler circuit has a
Hamilton circuit. The graph in Example 7 has a Hamilton
circuit but not an Euler circuit. The graph in Figure 23.19 has an
Euler circuit but not a Hamilton circuit.
F
A
C
E
M
Figure 23.19
Exercise 23D
SA
1 a Which of the following graphs has:
i an Euler circuit?
ii an Euler path but not an Euler circuit?
b Name the Euler circuits or paths found.
c
aA
B
C
A
bB
E
C
C
B
B
G
E
H
D
d
C
F
A
D
D
E
e
F
A
E
D
B
D
E
C
F
A
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Essential Further Mathematics – Module 5 Networks and decision mathematics
2 List a Hamilton circuit for each of the following.
a
A
B
C
E
D
b
B
C
c
A
F
C
H
I
D
A
D
G
B
E
F
E
e
D
B
B
A
E
F
G
E
D
A
F
E
C
d
C
B
PL
A
3 List a Hamilton path for this graph starting at
F and finishing at G.
F
E
G
H
D
Weighted graphs
M
23.5
C
A weighted graph is one where a number is associated with each edge. These numbers are
called weights.
SA
Examples of weighted graphs arise when the vertices of a graph are towns on a map and the
edges are the roads between the towns. The number assigned to each edge is the distance
between the towns represented by the vertices of that edge. Figure 23.20 is a weighted graph
representing towns and the roads connecting them.
Appleville
12
Snugvill
6
20
Melville
8
7
Cudstop
9
5
13 Clearview
11
Heavytown
Figure 23.20
The numbers in a weighted graph may also represent times, fuel consumption, cost, and so
on. The number attached to an edge is called the weight of that edge.
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Chapter 23 – Undirected graphs
627
Trees
A tree is a connected graph that contains no circuits.
Figure 23.21 is a tree. A tree has no multiple edges.
The simplest tree is a single vertex. A tree with n
vertices has n − 1 edges.
G
B
F
A
E
C
E
D
Figure 23.21
C
5
B
6
PL
Minimum spanning trees
A cable is needed to connect the communication systems
of the four towns in figure 23.22. It is convenient to lay the
cable alongside existing roads but it is not necessary to lay
the cable next to all roads.
A
4
3
5
D
6
Figure 23.22
M
A subgraph connecting all four towns can be chosen. It is
sensible for such a subgraph to be a tree (in red in figure 23.22)
shown in figure 23.23.
B
A
C
5
4
3
D
Figure 23.23
SA
A spanning tree is a subgraph that contains all the vertices of the original graph and is a tree.
A minimum spanning tree for a weighted graph is a spanning tree for which the sum of the
weights of the edges is as small as possible.
Figure 23.23 is a minimum spanning tree for figure 23.22.
Prim’s algorithm
Prim’s algorithm is a set of rules to determine a minimum spanning tree for a graph.
Examples of where Prim’s algorithm applies include such problems as the following
‘For a certain number of cities, a railroad network is to be developed and the
cost of connecting any two given cities is known. Find how all the cities
should be connected by rail to minimise total cost.’
Similar problems involve utility connections, e.g. water, gas, electricity.
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628
Essential Further Mathematics – Module 5 Networks and decision mathematics
Prim’s algorithm:
1 Choose a vertex and connect it to a second vertex chosen so that the weight of the edge
is as small as possible.
2 In each step thereafter, take the edge with the lowest weight, producing a tree with the
edges already selected. (If two edges have the same weight the choice can be arbitrary.)
3 Repeat until all the vertices are connected and then stop.
Example 8
E
Applying Prim’s algorithm
Apply Prim’s algorithm to obtain a minimum spanning
tree for the graph shown. Write down its weight, and
compare it to the weight of the original graph.
C
8
B
3
6
2
D 5
5
PL
A
7
6
6
E
2
F
Solution
Step 1
Step 2
B
2
Step 5
C
C
B
3
2
D
A
Step 4
5
C
B
B
2
3 5
M
A
5
A
A
Step 3
B
2
5
2
A
D
3 5
D
5
E
E
2
F
SA
The total weight is 17. The total weight of the original graph is 50.
Exercise 23E
1 Find a minimum spanning tree for each of the following graphs and give the total weight.
a A
2
6
B
2
b
3
C
2
F
5
1
D
16
16
3
10
16
E 17
17
A
C
24
B
E
2
12
G
17
11
D
15
F
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Chapter 23 – Undirected graphs
629
c
B
10
E
18
A
20
9
18
70
H
19
10
D
d
C
18
100
140
C
80
90
G
200
90
F
B
D
E
100
90
120
E
A
2 By trial and error find the shortest path from A to E.
B
6
4
F
7
D
8
A
3
5
PL
3
E
C
3 Find the shortest Hamilton path for the following graph,
starting at A.
B
5
C
6
3
G
6
4
4
A
D
2
5
F
3
4
5
E
SA
M
3
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Essential Further Mathematics – Module 5 Networks and decision mathematics
Key ideas and chapter summary
Graph
A graph consists of a set of elements
called vertices and a set of elements called
edges. Each edge joins two vertices
A figure such as the one drawn opposite is called
a graph.
F
B
A
C
E
E
D
The points A, B, C, D, E and F are called the vertices or nodes of the
graph and the lines connecting the vertices are called edges of the graph.
Adjacent vertices
The vertices A and B of a graph are adjacent vertices if they are joined
by an edge
Adjacency matrix
The adjacency matrix A of the graph
is an n × n matrix in which for example
the entry in row C and column F is the
number of edges joining vertices C
and F. For the graph above the
adjacency matrix is as shown.
Multiple edges
and loops
The graph is said to have multiple edges as there
are two edges joining A and D. C has one
edge, which links C to itself. This edge
is called a loop.
PL
Vertices (nodes)
and edges
A

A 0

B 1

C
0
D
0

E 1
F 1
B
1
0
1
0
0
1
C
0
1
0
1
1
0
M
Bipartite graph
D
0
0
1
0
0
1
C
E
1
0
1
0
0
0
F

1

1

0

1


0
0
A
D
B
A bipartite graph is a graph whose set of vertices can be split into two
subsets X and Y in such a way that each edge of the graph joins a vertex
in X and a vertex in Y.
SA
Review
630
Degree of a vertex
At each non-isolated vertex, A, in a graph there will be some edges
joined to A (adjacent with vertex A). The degree of vertex A, written as
deg(A) is the number of edges incident with it. Loops are counted twice.
Simple graph
A simple graph is a graph with no loops or multiple edges.
Degenerate graph
A graph is said to be degenerate if all its vertices are isolated.
Path
A path can be thought of as a sequence
of edges of the form AB, BG, GE, EC, CD,
DE, EG, GF
B
C
A
D
G
E
F
Connected graph
A graph is said to be connected if there is a path between each pair of
vertices.
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Chapter 23 – Undirected graphs
631
B
A circuit is a sequence of edges linking successive
vertices that starts and finishes at the same vertex. A
C
D
E
A subgraph of a graph consists of selected edges and vertices of the
graph with the same links as the original graph and with the selected
vertices including all the endpoints of the selected edges.
Planar graph
A graph that can be drawn in such a way that no two edges meet (or
have common points), except at the vertices where they are both
incident, is called a planar graph.
Euler’s formula
Euler’s formula states that for any connected planar graph,
v − e + f = 2, where v = the number of vertices,
e = the number of edges, and f = the number of faces.
Complete graph
A complete graph is a graph with edges connecting all pairs of vertices.
Euler path
A path that includes every edge just once is called an Euler path.
Euler circuit
An Euler circuit is an Euler path that starts and finishes at the same
vertex.
Hamilton path
A Hamilton path is a path through a graph that passes through each
vertex exactly once.
Hamilton circuit
A Hamilton circuit is a Hamilton path that starts and finishes at the
same vertex.
Weighted graphs
A weighted graph is a graph where a number is associated with each
edge. These numbers are called weights.
SA
M
PL
E
Subgraph
Tree
A tree is a connected graph that contains
no circuits.
A tree has no multiple edges. The simplest tree
is a single vertex. A tree with n vertices has
n − 1 edges.
Spanning tree
A spanning tree is a subgraph that contains all the vertices of the
original graph and is a tree.
Minimum spanning
tree
A minimum spanning tree for a weighted graph is a spanning tree for
which the sum of the weights of the edges is as small as possible.
Prim’s algorithm
Prim’s algorithm is a set of rules to determine a minimum spanning
tree for a graph.
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Circuit
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Essential Further Mathematics – Module 5 Networks and decision mathematics
Skills check
E
Having completed this chapter you should be able to:
construct an adjacency matrix from a graph and draw the graph from an adjacency
matrix
determine whether a graph is simple, connected, planar or complete
determine whether a graph has an Euler circuit or path
by trial and error, determine whether a graph has a Hamilton path or circuit
find a minimum spanning tree of a graph
PL
Multiple-choice questions
1 The minimum number of edges for a graph with seven vertices to be connected is:
A 4
B 5
C 6
D 7
E 21
2 Which of the following graphs is a spanning tree for the
network shown?
1
3
2
6
4
B 1
M
A 1
3
3
5
4
5
7
7
C 1
D 1
3
2
4
5
7
7
6
6
4
5
2
2
SA
Review
632
3
E 1
2
6
2
6
4
6
5
4
7
3 For the graph shown, which vertex has degree 5?
A Q
B T
C S
D R
E U
3
5
7
Q
S
P
U
R
T
4 A connected graph on 15 vertices divides the plane into 12 regions. The number of
edges connecting the vertices in this graph will be:
A 15
B 23
C 24
D 25
E 27
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Chapter 23 – Undirected graphs
633
Review
5 Which of the following graphs does not have an Euler circuit?
C
A
B
D
PL
E
E
6 A connected planar graph divides the plane into a number of regions. If the graph
has eight vertices and these are linked by 13 edges, then the number of regions is:
A 5
B 6
C 7
D 8
E 10
A
B
F
C
M
7 For the graph shown, which of the following paths is a Hamilton
circuit?
E
A ABCDCFDEFAEA
B AEFDCBA
C AFCDEABA
D
D ABCDEA
E AEDCBAF
SA
8 For which one of the following graphs is the sum of the degrees of the vertices
equal to 12?
C
B
A
D
E
9 The sum of the degrees of the vertices on the graph shown
here is:
A 20
B 21
C 22
D 23
E 24
10 For the graph shown, which additional arc could be added
to the network so that the graph formed would contain an
A
Euler path?
A AF
B DE
C AB
D CF
E BF
B
C
E
F
D
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Essential Further Mathematics – Module 5 Networks and decision mathematics
11 For the graph shown here, the minimum length spanning
tree has length:
A 30
B 31
C 33
D 34
E 26
7
8
4
5
4
10
6
9
2
E
12 Of the following graphs, which one has both Euler and
Hamilton circuits?
C
B
PL
A
D
E
13 Which one of the following graphs has an Euler circuit?
C
B
M
A
D
E
SA
Review
634
2
14 Which one of the following is a spanning tree for
the graph shown here?
A
3
1
5
B
2
2
3
1
4
3
1
5
4
5
4
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Chapter 23 – Undirected graphs
635
D
2
E
2
2
3
3
1
1
1
4
5
5
5
4
E
15 Which one of the following graphs has an Euler circuit?
B
A
D
PL
C
E
M
16 Which one of the following graphs provides a counter-example to the statement:
‘For a graph with seven vertices, if the degree of each vertex is greater than 2 then
the graph contains a Hamilton circuit’?
C
B
A
E
SA
D
17 For which one of the following graphs is the sum of the degrees of the vertices equal
to 20?
A
B
C
D
E
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C
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Essential Further Mathematics – Module 5 Networks and decision mathematics
18 Which one of the following paths is a Hamilton
circuit for the graph shown here?
A PQRSTP
B PQRSTUVP
C PQUVRSTP
D PQRSTUVUTP
E PQRSTUVRVQUTP
Q
R
V
U
P
S
T
B
E
19 Four towns, A, B, C and D are linked by roads as shown.
Which of the following graphs could be used to
represent the network of roads? Each edge
represents a route between two towns
C
A
B
PL
A
B
C
B
C
A
D
B
D
B
C
C
A
A
D
D
D
E
B
C
C
A
A
M
D
D
Extended-response questions
1 This question is about the vertices of a graph and the degree of a vertex. In Graph A
below, there are four vertices (the dots).
SA
Review
636
Graph A
Graph B
1
2
3
4
Graph C
a Complete the table for Graph B.
Degree
0 1 2 3 4 5 6 7
b Study Graphs A, B and C and then
Number of vertices
consider the statement:
‘In any graph the total number of vertices of odd degree is an even number.’
Is this statement true for Graphs A, B and C? How many vertices of odd degree
does each graph have?
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Chapter 23 – Undirected graphs
637
Review
c To Graph B an extra edge is added joining two vertices of even degree. Does the
above statement still hold true for this graph? Explain the reasons for your answer.
d From Graph C an edge joining an odd degree vertex to an even degree vertex is
removed. Does the above statement still hold true for this graph? Explain the
reasons for your answer.
a Explain why this is a planar graph.
b For this graph, write down:
i the number of vertices
ii the number of edges
iii the number of faces
E
2 Consider the graph shown here.
PL
Note: the region outside the graph is counted as a face.
c Draw a spanning tree for this graph.
d For the spanning tree drawn in c, write down
i the number of vertices
ii the number of edges
iii the number of faces
M
3 Let H denote a planar graph with n vertices.
a If T denotes a spanning tree of H, specify:
i the number of vertices in T
ii the number of edges in T
iii the number of faces in T
b Hence verify that, for the graph T:
number of vertices − number of edges + number of faces = 2.
SA
4 The map shows six campsites, A, B, C, D, E
and F which are joined by paths. The numbers
by the paths show lengths in kilometres of
sections of the paths.
a
6
B
9
lake
5
A
3 D
4
1
5
2
F
3
4
C
5
E
i Complete the graph opposite which shows
B
9
6
the shortest direct distances between
4
campsites. (The campsites are represented
C
A
8
D
by vertices and paths are represented by
5 F 7
edges.)
4
5
ii A telephone cable is to be laid to enable each
8
E
campsite to phone each other campsite. For
environmental reasons, it is necessary to lay the cable along the existing paths.
What is the minimum length of cable necessary to complete this task?
(cont’d.)
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Essential Further Mathematics – Module 5 Networks and decision mathematics
A

A 0

B 1

C
0
D
1

E 1
F 1
B
1
0
1
0
0
0
C D E F

0 1 1 1

1 0 0 0

0 1 1 0

− − − −


− − − −
− − − −
E
iii Fill in the missing entries for the
matrix shown for the completed
graph formed above.
M
PL
b A walker follows the route A−B−A−F−E−D−C−E−F−A.
i How far does this person walk?
ii Why is the route not a Hamilton circuit?
iii Write down a route that a walker could follow which is a Hamilton circuit.
iv Find the distance walked in following this Hamilton circuit.
c It is impossible to start at A and return to A by going along each path exactly
once. An extra path joining two campsites can be constructed so that this is
possible. Which two campsites need to be joined by a path to make this possible?
SA
Review
638
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