Review for Exam #2

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Review
Chapter 5
1a)Draw two different representations of a graph with edge sets E={AB,AC,AD,AD,CC}
and vertices V={A,B,C,D,E}
1b) Give the degree of every vertice in part 1a)
1c) List all edges adjacent to AB
1d) Is the graph connected or disconnected?
2) In the following graph
a) Determine if an Euler circuit, Euler path, or neither exist.
b) If there is an Euler path or circuit, where should it start and stop?
3) True or false?
a) The Euler Circuit theorem says that if the degree of all the vertices in a graph are odd, then the graph has an
Euler circuit.
T
F
b) The Euler path theorem says that is there are two vertices of odd degree, then an Euler path exists. T
F
c) It is possible that the sum of all degrees of all vertices of a graph to be an even number.
T
F
4) How many edges are there in a graph with vertices of degree 2,3,3,3,2,3?
5) Use Fleury’s Algorithm to find an Euler path or circuit in the following graph(label edges 1,2, etc):
6) Find an optimal Eulerization of the following graph
Chapter 6
1) For the following graph
A
E
C
G
D
F
B
a) find a Hamiltonian circuit
b) find a Hamiltonian path that begins at A and ends at B
2a) Draw the complete graphs K6,K5,K4, and K3.
b) Find the number of edges and Hamiltonian circuits in the above graphs.
3) In the following graph, find the most efficient Hamiltonian circuit by using
B
2
A
5
6
3
C
4
2
D
a) Brute force
b) Nearest neighbor algorithm
c) What is the error in the nearest neighbor algorithm?
d) Repetitive nearest neighbor algorithm.
Chapter 7
1) Let T represent tree and N represent network. Circle one, both, or neither T or N in each of the following graphs.
T
N
T
N
2) True or False?
a) In a tree, every edge is a bridge
T
F
b) A tree with N-1 vertices has N edges T
F
c) If every edge of a graph is a bridge, then the graph is a tree.
T
d) A tree must be a connected graph
T
F
e) Every tree is a network
T
F
f) A spanning tree is a subgraph with no paths
T
F
3) Find the number of possible spanning trees for the following graph.
T
N
F
4) Use Kruskal’s algorithm to the minimum spanning tree in the following weighted network.
2.4
2.2
2.3
2.8
3.5
1.2
2.1
2.6
3.1
1.4
5) Assume that one of the three points- X,Y, or Z- is a Steiner point of triangle ABC. The distances of each of these
three points to the vertices of the triangle are given in the table. Determine which of the three points is the Steiner
point. Explain your answer.
X
54
61
125
A
B
C
Y
72
94
77
Z
41
87
104
6) Below is an equilateral triangle each side of length 1. Construct a 30-60-90 triangle with the length of the sides
labeled appropriately (split triangle in half, and use pythagorean theorem).
60
1
1
60
60
1
Chapter 8
1) For the digraph shown in the figure,
A
B
C
E
D
a) find all vertices that are incident to A.
b) find all vertices that are incident from A.
c) find all arcs adjacent to AC.
d) find the indegree of C
e) find the outdegree of C
2) #1 from text of Chapter 8
3) #28 from Chapter 8 in text
Also, review all quiz question (even though some of them are in this review) and all HW questions for
the exam. Have a nice spring break, and come back on Monday with questions!
I’ll post solutions to this review on Monday after the break.
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