Graph Theory Word Maze
There are 20 words associated with Graph Theory hidden in the square below.
The Start and Finish squares are indicated and the words lie in a continuous path from one to the other. All
moves are horizontal or vertical (not diagonal.).
The initial letter of each word and the order in which they appear are given to stop you getting lost!
Start
A
E
D
D
E
G
R
E
N
E
C
R
C
G
E
A
R
T
E
N
O
T
1.
A
2.
E
3.
P
4.
C
5.
T
6.
D
7.
L
8.
T
9.
A
10.
B
11.
R
12.
W
13.
D
14.
C
15.
V
16.
C
17.
E
18.
N
19.
S
20.
H
L
P
E
E
I
L
P
L
X
C
E
A
C
L
R
D
A
O
O
E
E
D
N
Y
E
T
J
L
E
R
T
U
L
E
C
E
C
A
P
T
E
O
N
E
B
T
N
A
L
M
E
V
D
N
R
I
O
U
W
K
O
I
S
E
A
I
P
R
T
E
D
C
M
A
M
I
L
A
E
T
G
I
H
P
H
N
O
T
R
T
I
R
A
P
L
E
I
A
N
Finish
Solution
Start
A
E
D
D
E
G
R
E
N
E
C
R
C
G
E
A
R
T
E
N
O
T
L
P
E
E
I
L
P
L
X
C
E
A
C
L
R
D
A
O
O
E
E
D
N
Y
E
T
J
L
E
R
T
U
L
E
C
E
C
A
P
T
E
O
N
E
1.
A
R
C
2.
E
D
G
E
3.
P
L
A
N
E
4.
C
Y
C
L
E
5.
T
R
E
E
6.
D
E
G
R
7.
L
O
O
P
8.
T
R
A
I
L
9.
A
D
J
A
C
E
N
T
10.
B
I
P
A
R
T
I
T
11.
R
O
U
T
E
12.
W
A
L
K
13.
D
I
G
R
A
P
H
14.
C
O
M
P
L
E
T
E
15.
V
E
R
T
E
X
16.
C
O
N
N
E
C
T
E
17.
E
U
L
E
R
I
A
N
18.
N
O
D
E
19.
S
I
M
P
L
E
20.
H
A
M
I
L
T
O
N
E
B
T
N
A
L
M
E
V
D
N
R
I
O
U
W
K
O
I
S
E
A
I
P
R
T
E
D
C
M
A
M
I
L
A
E
T
G
I
H
P
H
N
O
T
R
T
I
R
A
P
L
E
I
A
N
Finish
E
E
D
I
A
N
ted
a
subset of the vertices
and edges of larger graph
Line from vertex to vertex
a
oci
s
s
ra
be edge
m
u
e n th an
h
T
wi
een
etw
ts b
xis ces
he
i
pat f vert
o
e
her pair
any
G
ian
ler
Eu
h
rap
Bipartite Graph
re
the than o
s am n e
e p e dge
air
of betw
ver
e
tic en
es
... can be drawn without
edges crossing
Mo
... w
i
S em
−
a
No
de
Simple Graph
Va
lu
A
Trail with no
repeated vertices
no
ith es
g
lk w
wa ed ed
eat
rep
um
m
i
n
A
e
Point where edges meet
Mi
nin
n
a
Sp
ree
T
g
Wa
lk
A
rk
o
tw
e
N
5
D
h
p
a
igr
Gr
a
dir ph w
it
ect
ion h
al
edg
es
K
nec
tio
ns
... t
r
edg avers
e
ee
xac s eve
r
tly
on y
ce
V
x
e
t
er
Complete Graph
Gr
aph
of v whe
re
er
by t ice s t wo
an
ar e dis t
y
c
o m not inct s
mo
joi
et s
n
e dg ned
es
4c
on
Path
Eulerian Cycle
rte
Ve
ith
xw
Trail
ap
Gr
hw
odd
ith
ree
deg
c
no
ycl
... connects all vertices with
the least total edge length
dge
cle
Co
n
nec
w
i
t
h n ted g
o
od raph
d
n
ode
s
he
Odd node
Arc
wit
Cy
e
e
r
T
Weight
Complete graph
with five vertices
No
de
N
ode
Co
n
one necte
d
pa
ir o Grap
hw
fo
dd
ver ith
tic
es
es
h
wit
s
h
s
p
gra ection
d
pe
nn
sha x co
t
en
rte
fer e ve
f
i
D sam
the
ian
aph
Gr
... v
ler
Eu
Planar Graph
Va
le
n
c
y
De
g
re
e
of
ve
r
t
ex
ed
t
a
epe
e
g
d
e
Gr
a
of a ph co
ns
rc s
an isting
dn
od
es
isit
s
exa ever
y
ctl
y o verte
nc e
x
Isomorphic Graphs
th
ncy
pa
a
ed
e
Point where edges meet
R
Va
le
Connected Graph
of
x
rte
e
V
e
r
g
de
e4
s
clo
Sub
h
p
a
a
gr
...
g
Ed
sequence of
consecutive edges
.. . h
as
o
r
rep no lo
o
e
a
ted ps
e
dg
es
A
... where every vertex is
joined to every other vertex
Hamiltonian Cycle
Ed
ge
Cy
cle
ian
...
Graph where two distinct sets
of vertices are not joined
by any common edges
hpar Getit r api B
e gde det ae pe R
me
et
edg
es
wh
ere
Po
int
ree
V
T
ng
nn i
S pa
t ex
ver
No
de
of
nc y
e
l
a
e
gre
m
imu
Min
De
Pla
na
r
Gr
ap
h
a
h
wit
c es gth
n
e rti
ll v ge le
a
cts al e d
ne
t
con ast to
...
le
the
egde na hti w
det ai coss a r eb mun eh T
ph
gr a es
te
ple ver tic
m
Co five
with
f ro
rap
h
e
Lin
Ed
ge
x
erte
ov
t
ex
an
G
es
ycl
e rt
mv
K
More than one edge between
the same pair of vertices
can
b
e dg e dra
w
es
c ro n wit
ssi
ho
ng
ut
. ..
h
or as no
rep
l
ea o o ps
ted
ed
ges
s
tice a ph
ver r g r
the la rge
of
t
se
of
su b d g es
e
a nd
s e g de l a n oit ceri d
hti w h p ar G
Arc
E
ule
ri
c
no
ith
Co
n
on ne c te
ep
ai r d Gr
a
of
od ph w
dv
it
e rti h
ce s
−
ge
Ed
S em
i
hw
ap
Gr
Co
n
n
with ecte
d
n
o o gr ap
dd
h
nod
es
e
Tre
Digraph
s egde evi t uces noc
f o ecneuqes A
Weight
...
Eu
l er
ath
Co
nn
ec t
ed
Gr
aph
d
No
e
alu
V
e
Pa
th
e
gre
de
f
xo
r te
Ve
h
A
Walk
li ar T
Sim
a
p
l
e
Gr
ap
on s
cti
h
rap
bg
Su
...
a
A walk with no
repeated edges
ec no ylt c axe
xet r ev yr eve sti si v...
ler
ian
Gr
aph
xet r ev r e ht o yr eve ot de ni oj
si xet r ev yr eve er e h w...
wh
er e
an
y p pa th
air
exi
sts
of
ver b e
twe
tic
en
es
ne
con
cy
len
Va
e
edg
T
rep rail
w
e
a
t
ed ith no
ver
tic
es
h
wit
A
Hamiltonian Cycle
Eu
p
sed
clo
rk
Ne
two
x
rte
Ve
et
me
.. .
tra
edg ve rs
es
ee
e
x
a
ctly v ery
on
ce
g es
ed
Gr
ap
of a h co
ns
rc
s
a nd isting
no
des
re
he
s noi t ce nnoc xet r ev e mas e ht
ht i ws hpar g de pahs t ner ef f i D
de
No
4
Complete Graph
Isomorphic Graphs
eer ge d ddo hti w xet r e V
4
cle
Cy
w
int
Po
Odd node
a
5
Bin Packing Exercise
Where objects of varying sizes must be placed into containers of a fixed
capacity, the problem is described as ‘bin packing’. The name is used for any
problem of this general type, whether to do with objects, lengths, times or
whatever the scenario.
Two algorithms are commonly used to attempt to solve these problems:
First-Fit Bin Packing:
Number the bins, then always place the next item in the lowest numbered
bin which can take that item.
First-Fit Decreasing Bin Packing:
Reorder the items into decreasing order of size.
Number the bins, then always place the next item in the lowest numbered
bin which can take that item.
Activity
A builder uses piping of standard length 12 metres.
The following sections of varying lengths are required for a particular job:
Section
Length
(metres)
A
B
C
D
E
F
G
H
I
J
K
L
2
2
3
3
3
3
4
4
4
6
7
7
Explore how many standard 12 m lengths of pipe will be required if each of the
following methods is used:
(a) First-fit bin packing
(b) first-fit decreasing bin packing
(c) trial and improvement
[Adapted from p. 209-210, AEB Discrete Mathematics, Heinemann, 1992]
Length of standard pipe section (metres)
Length of standard pipe section (metres)
Bins
12
11
10
9
8
7
6
5
4
3
2
1
12
11
10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
Pipe Sections
A(2)
E(3)
B(2)
C(3)
D(3)
K(7)
L(7)
K(7)
L(7)
F(3)
G(4)
J(6)
H(4)
I(4)
A(2)
B(2)
E(3)
C(3)
D(3)
F(3)
G(4)
J(6)
H(4)
I(4)
Solutions
(a) First-fit bin packing
12
Length of standard pipe section (metres)
11
10
9
D(3)
8
G(4)
7
6
I(4)
C(3)
5
F(3)
4
3
B(2)
2
1
A(2)
1
J(6)
E(3)
2
K(7)
L(7)
5
6
7
8
6
7
8
H(4)
3
4
(b) First-fit decreasing bin packing
12
A(2)
Length of standard pipe section (metres)
11
F(3)
10
9
G(4)
H(4)
I(4)
8
E(3)
7
6
5
4
D(3)
K(7)
L(7)
3
J(6)
2
C(3)
1
1
2
3
4
B(2)
5
(c) Trial and improvement
12
Length of standard pipe section (metres)
11
A(2)
B(2)
I(4)
10
9
C(3)
F(3)
D(3)
8
E(3)
7
H(4)
6
5
4
K(7)
L(7)
J(6)
3
G(4)
2
1
1
2
3
4
5
6
7
8
Sprouts
This is a simple pen-and-paper game for two players which involves arcs and
nodes. It was invented in 1967 by Professor John H Conway and Michael S
Paterson and has been described and analysed in books and on the internet.
(Enter the words sprouts and Conway into a search engine for more information.)
Sprouts with three spots:
Draw three spots anywhere on the paper.
Each player in turn draws a line joining one spot to another spot (or itself) and
places a new spot somewhere on this line.
No lines may cross,
No spot may have more than three lines coming out of it (i.e. we would say that
the degree of any vertex cannot exceed three).
The game continues until no further moves are possible and the player to make
the final move is the winner.
A possible three spot game:
Analysis:
Providing students have met the fact that for any graph the node-sum is twice
the number of edges (from the hand-shake lemma), then it is quite easy to prove
that a three spot game of sprouts can never exceed 8 moves.
The game begins with 3 vertices and no edges.
Every move adds to the network 2 edges and one vertex.
Therefore after m moves, the network will have gained 2m edges and m vertices.
So the final graph contains 2m edges and m + 3 vertices.
Since the winning move will leave at least one vertex of degree 2, the maximum
number of 3-nodes is m + 2.
Therefore
node sum
(m + 2) x 3 + 2
and since
node sum = 2 x (no. of edges)
we have
so
and so
2 x (2m)
4m
m
(m + 2) x 3 + 2
3m + 8
8
Thus no game of three spot sprouts can exceed 8 moves.
Sprouts can be played with any number of spots and it can be shown that an n
spot game will never exceed 3n – 1 moves.
Brussels Sprouts:
A similar game can be played with crosses instead of spots, where each cross
represents a vertex of maximum degree 4. A move joins two ‘branches’ of
existing crosses (or a cross to itself) and a cross is placed on the new line.
The start of a possible two cross game:
In fact in Brussels Sprouts the first player always wins if the number of crosses
is odd and the second player always wins where the number of crosses is even!