5.1
/1-8, 16-18, 37-40
1.
Give the function
g
that is part of the formal definition of the directed graph:
g
(
a
) = (1, 2),
g
(
b
) = 1, 3),
g
(
c
) = (2, 3),
g
(
d
) = (2, 2)
2. Answer the following questions about the accompanying graph
a. Is the graph simple?
yes
b. Is the graph complete?
no.
e.g. – no arc between 1 and 3
c. Is the graph connected?
yes.
there is a
path
between every two nodes
d. Find two paths from 3 to 6.
a
3
, a
4
, a
6
a
5
, a
6
e. Find a cycle.
a
3
, a
4
, a
5
f. Find an arc whose removal will make the graph acyclic.
a
5
g. Find an arc whose removal will make the graph not connected.
a
6
3.
Sketch a picture of each of the following graphs.
a. A simple graph with three nodes, each of degree 2
b. graph with 4 nodes, with cycles of length 1, 2, 3, 4
c. noncomplete graph with four nodes, each of degree 4
4.
a. Which nodes are reachable from node 3?
3,
4, 5 and 6
b. What is the length of the shortest path from 3 to 6?
2 (3-5-6)
c. What is a path from node 1 to node 6 of length 8?
(1-2-1-2-1-2-2-1-6)
5. a. Draw
K
6.
b. Draw
K
3,4
c
a
b
a
1
2
3
b
c
1
2
3
4
5
6
7
a
1
a
2
a
3
a
4
a
5
a
6
a
7
2
3
5
6
1
d
4
6.
For each of the following characteristics, draw a graph or explain why such a graph does not exist.
a. four nodes of degree 1, 2, 3, and 4, respectively.
b. simple, four nodes of degree 1, 2, 3, and 4, respectively.
Since we do not allow loops or parallel arcs, the
degree of each node has to be obtained by the number of other nodes it is connected to.
Therefore no node can
have a degree of four since for each node there are only three other nodes.