REVISION ON UNDIRECTED GRAPHS
QUESTIONS
Multiple-choice questions
1 The minimum number of edges for a graph with seven vertices to be connected is:
A4
B5
C6
D7
E 21
2 Which of the following graphs is a spanning tree for the network shown?
3 For the graph shown, which vertex has degree 5?
3 For the graph shown, which vertex has degree 5?
AQ
BT
CS
DR
EU
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
5 Which of the following graphs does not have an Euler circuit?
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:
A5
B6
C7
D8
E 10
7 For the graph shown, which of the following paths is a Hamilton circuit?
A ABCDCFDEFAEA
C AFCDEABA
E AEDCBAF
B AEFDCBA
D ABCDEA
8 For which one of the following graphs is the sum of the degrees of the vertices equal to
12?
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 Euler path?
A AF
E BF
B DE
C AB
11 For the graph shown here, the minimum length spanning tree has length:
A 30
B 31
C 33
D 34
E 26
12 Of the following graphs, which one has both Euler and Hamilton circuits?
D CF
13 Which one of the following graphs has an Euler circuit?
14 Which one of the following is a spanning tree for the graph shown here?
15 Which one of the following graphs has an Euler circuit?
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’?
17 For which one of the following graphs is the sum of the degrees of the vertices equal
to 20?
18 Which one of the following paths is a Hamilton circuit for the graph shown here?
A PQRSTP
B PQRSTUVP
E PQRSTUVRVQUTP
C PQUVRSTP
D PQRSTUVUTP
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.
Extended- response questions
1 Consider the graph shown here.
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
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
2 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.
3 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 i Complete the graph below which shows the shortest direct distances between
campsites. (The campsites are represented by vertices and paths are represented by
edges.)
ii A telephone cable is to be laid to enable each 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?
iii Fill in the missing entries for the matrix shown for the completed graph formed above.
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?
ANSWERS
Multiple-choice questions
1C
2C
3A
4D
10B
11A
12A
13E
19B
5A
14C
Extended – Response
1 b iThe number of vertices = 5
ii The number of edges = 8
iii The number of faces = 5
di The number of vertices = 5
ii The number of edges = 4
iii The number of faces = 1
2a I n vertices ii n −1edges iii one face
b n +1−(n−1)=2
3a I add arc CE
Length =24km
Iii 1 0 1 1
1101
0110
b i 45km ii Repeated visits to sites
iii ABCDEFA iv 33 km
cF&C
6C
15B
7B
16D
8C
17A
9C
18C