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Faculty of Mathematics Centre for Education in
Waterloo, Ontario N2L 3G1 Mathematics and Computing
Grade 6 Math CirclesFebruary 11/12, 2014
Graph Theory I- Solutions“*” indicates challenge question
1. Trace the following walks on the graph below. For each one,
state whether it is a path?
How do you know? (b) and (d) are paths since they do not repeat
vertices. Notice
that A is repeated in (a) and (c), and E is repeated in (e).
(a) L-C-E-A-B-A-D
AB
C
DE
FG
HI
J
K
L
(b) H-F-G-J-D-A
AB
C
DE
FG
HI
J
K
L
(c) F-D-A-B-K-E-A
AB
C
DE
FG
HI
J
K
L
(d) F-G-J-H
AB
C
DE
FG
HI
J
K
L
(e) D-A-E-B-K-E-C
AB
C
DE
FG
HI
J
K
L
1
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2. Find 10 walks from 1 to 4 in the following graph. How many of
these walks are paths?
How many possible paths are there between 1 and 4. (Hint: You
may want to redraw
the graph)
1
2
3
4
56
7
8
910
11
There are many walks possible. Students should strive to find at
least 10. Remember
that walks can include repeated vertices. There are 8 paths from
1 to 4: 1-4, 1-2-4,
1-2-5-8-3-10-4, 1-2-10-4, 1-6-8-3-10-4, 1-6-8-3-10-2-4,
1-6-8-5-2-4, 1-6-8-5-2-10-4.
3. Find all the cycles in the following graph. There are 6
cycles: 4-8-5-1-4, 1-2-6-5-1,
5-6-9-5, 1-2-6-9-5-1, 4-1-2-6-5-8-4, 1-2-6-9-5-8-4-1.1 2
4 5 6
89
4. A complete graph is a graph that has all possible edges. This
means that every
vertex is connected to every other vertex. We name a graph of
this type Kn where n
is the number of vertices.K5 is an example of this type of
graph. Which edges should
be added to the graph below to make it K5? Write the names of
the edges and then
draw them on the graph. The edges {a,b}, {a,d}, and {e,b} should
be added so thegraph looks like the one below.
a
b
cd
e
2
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5. A k-regular graph is a graph with k edges incident
to each vertex. An example of this is the Petersen
Graph which is a type of 3-regular graph.
Luc, Ryan, Vince, Emily, and Nadine go to a party.
When they get there they want to shake hands but
they only have time to shake two other people’s
hands. Draw two different graphs to show how this
can happen. How many handshakes are there? Do
the rules for drawing a graph make sense in this sit-
uation?
a
b
cd
e
f
g
h
i
j
Petersen GraphGraphs will vary. Each vertex should be incident
to 2 edges. There should be 5 edges
total. There are 5 handshakes. Yes, the rules make sense since
you can not shake hands
with yourself and it does not make sense to have two different
handshakes between the
same people since they have already shaken hands once.
6. *Is a vertex always on the same level if the BFST is created
with a different root? No,
the level of a vertex may change depending on the root that is
used to create it.
7. *The length of a path is the number of edges in the path. For
example, for a path
1-2-3-4, the length would be 3. The shortest path is the path
with the least number
of edges possible. What does the level of a vertex in a BFST
tell you about the
relationship between the root and a vertex? The level of a
vertex is the length of the
shortest path between the root and the vertex.
8. *How can you use a BFST to determine the shortest path
between two vertices? Create
a BFST with one of the vertices as the root. The level of the
other vertex is the length
of the shortest path.
9. **Explain why a BFST gives the shortest path between the root
and any other vertex.
(Hint: think about where the vertex would be if there was a
shorter path) A BFST
gives the shortest path because if there was a shorter path the
vertex would be higher
in the tree. Remember that a vertex is drawn on the tree as soon
as it is connected to
a vertex that is already in the tree.
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10. **Does rearranging a BFST create another valid BFST? Either
explain why it does or
show that it doesn’t by creating a BFST and rearranging it.
(Hint: think about the
order vertices are drawn on the tree. Would it change if you
started with a different
root?) No, it does not create another BFST. Although it is still
a tree, it may not be
a BFST since the order in which vertices are connected in the
tree will change. For
example, try drawing a graph where two vertices are on the same
level in a BFST and
are adjacent in the graph. Notice how these two vertices are not
adjacent in the BFST.
Now try drawing a BFST with one of these vertices as the root.
These two vertices
should now be adjacent. This would not have happened if you
tried to rearrange the
other BFST.
11. Find a shortest path between 1 and 4. Check your answer by
drawing a BFST. Answers
will vary. The shortest path has a length of 3.
1
23
4
5
6 7
8
9
10
12. Below is a graph of colours from an imaginary world. An edge
between two vertices
means that you can create those colours from each other. What is
the fastest way to
create yellow if you are starting with green? Use a BFST to
solve. BFST’s will vary.
The shortest path has length 4.
blue
purple
red
orange
yellow
greenpink
teal
4
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13. Sarah has a letter that she wants to give to Emily. But they
won’t see each other
because they are in different cities. Below is a map of people
who will see each other.
Use a BFST to find the quickest way for Sarah to get the letter
to Emily and then
answer the questions. The quickest way is the following path:
Sarah-Vince-Ryan-Luc-
Nadine-Emily
Emily
Sarah
Tim
Ryan
Luc
Vince
Nadine
Ishi
Kamil
Lawren
Cass Dalton
(a) Which people must see each other in order for Sarah to get
the letter to Emily?
In other words, if these people don’t meet then it is impossible
for Sarah to get
the letter to Emily. Ryan and Vince, Luc and Nadine, and Nadine
and Emily
must meet.
(b) What do these vertices have in common? They are all
bridges.
(c) If Ryan and Vince don’t meet, what changes about the graph
(other than removing
an edge)? The graph is not connected. There are now 2
components.
(d) Vince lives far from Sarah so she doesn’t want to give him
the letter. What is the
fastest way now?
Sarah-Tim-Lawren-Dalton-Cass-Vince-Ryan-Luc-Nadine-Emily
(e) Dalton has a letter that he wants to give to Ishi. What is
the fastest way for him
to do this? Dalton-Cass-Vince-Ryan-Kamil or Luc-Ishi
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