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