Ali Mahmudi, Introduction to Graph Theory [5]
Example 1.1
Let G = VG, EG is a graph, where VG = {1, 2, 3, 4} and EG = {1,2, 1,3, 1,4, 2,4}. We can represent the graph as follow.
Figure 1.4. A Graph with Four Vertices and Five Edges
C. Graph as a Model
Graph theory provides a natural mathematical model for some problems. At this chapter we only describe the problem and delay detailed discussion of their
solution to later chapters.
Problem 1
Suppose that the graph of Figure 1.5 represents a network of telephone lines and poles. We are interested in the networks vulnerability to accidental disruption. We
want to identify those lines and poles that must stay in service to avoid disconnecting the network.
Figure 1.5. Network of Telephone Line 2
1
4 3
D A
B
C E
F e
1
e
2
e
3
e
4
e
5
e
6
e
7
e
8
Ali Mahmudi, Introduction to Graph Theory [6]
There is no single line whose disruption removal will disconnect the graph network, but the graph will become disconnected if we remove the two lines
represented by edges e
4
and e
5
, for example. When it comes to poles, the network is more vulnerable since there is a single vertex, vertex d, whose removal
disconnects the graph. This illustrates the notions of edge connectivity and vertex connectivity of a graph.We may also want to find a smallest possible set of edges
needed to connect the six vertices. There are several examples of such minimal sets. One is {e
1
, e
3
, e
5
, e
6
, e
7
}
Problem 2
Suppose we have three houses each of which have to be supplied with electricity, gas and water. Is it possible to connect each utility with each of the three houses
without the lines or mains crossing? We can represent the connection of the three houses to the three utilities by
the graph of Figure 1.6. Here we have a graph with six vertices, three of which represent the houses denoted by H
1
, H
2
, H
3
, the other three represent the utilities denoted by E, G, W, and an edge joins two vertices if and only if one vertex
denotes a house and the other vertex a utility.
Figure 1.6 The Three Utilities Graph The problem then is as the whether or not we can draw this graph in such a
way that no two edges intersect. The answer is no – we will see why when we look at planar graphs in Chapter 4.
E H
1
H
2
H
3
G W
Ali Mahmudi, Introduction to Graph Theory [7]
Problem 3
A company has branches in each of six cities C
1
, C
2
, C
3
, …, C
6
. The airfare for a direct flight from C
i
to C
j
is given by the i, j
th
entry of the following matrix where
indicates that there is no direct flight. For example the fare from C
1
to C
4
is IDR 400.000,00 from C
2
to C
3
is IDR 1.500.000,00.
55 25
25 10
55 10
20 25
25 10
10 20
40 20
10 15
25 20
15 50
10 25
40 50
The company is interested in computing a table of cheapest fares between pairs of cities. Even if there is a direct flight between two cities this may not be
the cheapest route. We can first represent the situation by a weighted graph, i.e., a graph with “weights” attached to edges according to the airfares, as in Figure 1.7.
Figure 1.7 The Weighted Graph of Airfare for Direct Flights Between Six Cities The problem can then be solved using Dijkstra’s algorithm. This type of
problem is called a shortest path problem. 50
A
B
C
D E
F 15
10 10
55 10
25 25
20 20
25 40
Ali Mahmudi, Introduction to Graph Theory [8]
D. Some Type of Graphs
