Graph Theory
Name:
Study Guide
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Date:
Define each of the following. It may be helpful to draw examples that illustrate the vocab word and/or
counterexamples to define the word.
1. Graph ~
2. Vertex ~
3. Edge ~
4. Connected Graph ~
5. Non-connected Graph ~
6. Path ~
7. Circuit ~
8. Euler Path ~
9. Euler Circuit ~
10. Valence ~
11. complete graph ~
12. Eulerize a Graph ~
13. Hamiltonian Circuit ~
14. algorithm ~
15. Brute Force Method ~
16. method of trees ~
17. fundamental counting principle ~
18. Nearest – Neighbor Algorithm ~
19. Sorted – Edges Algorithm ~
20. Traveling Salesman Problem ~
21. Chromatic Number ~
22. Four-color map problem ~
CHAPTER 1:
1. What is the valence of vertex A in the graph below?
2. The valences of the vertices in the accompanying
graph listed in decreasing order are
(a) 3
(b) 2
(a) 1, 1, 1, 2, 3, 3, 4, 5
(c) 4
(b) 5, 4, 4, 2, 1, 1, 1.
(c) 5, 4, 3, 3, 2, 1, 1, 1.
3. The graph below is not connected because
4. A vertex in the graph below that is even-valent is
(a) it has all even-valent vertices.
(a) C.
(b) it consists of two parts.
(b) A.
(c) E.
(c) it consists of three parts.
5. The graph below has
6. Which of the following statements is true about a
path?
(a) A path always forms a circuit.
(a) four vertices and six edges.
(b) A path is always connected.
(b) four vertices and four edges.
(c) A path can visit any vertex only once.
(c) five vertices and six edges.
7. If a graph consists of four vertices and every pair of
8. It is not possible for a graph to have five vertices of
vertices is connected by a single edge, how many edges
valence 3 and six vertices of valence 4 because
are in the graph?
(a) there are no graphs with exactly 11 vertices.
(b) a graph cannot have an even number of 4-valent
(a) four
(b) five
(c) six
vertices.
(c) a graph cannot have an odd number of odd-valent
vertices.
9. If a graph is connected and has six vertices, what
10. Consider the path represented by the sequence of
can be said about the number of edges in the graph?
numbered edges on the graph below. Which statement
is correct?
(a) There are at least five edges in the graph.
(b) There are exactly five edges in the graph.
(c) There are at least six edges in the graph.
(a) The sequence of numbered edges forms an Euler
circuit.
(b) The sequence of numbered edges traverses each
edge exactly once but is not an Euler circuit.
(c) The sequence of numbered edges forms a circuit.
11. For the graph below, which statement is correct?
12. Suppose each vertex of a graph represents a
baseball team and each edge represents a game played
by two baseball teams. If the resulting graph is not
connected, which of the following statements must be
(a) The graph has an Euler circuit.
true?
(b) One new edge is required to eulerize the graph.
(a) At least one team never played a game.
(c) Three new edges are required to eulerize the graph.
(b) At least one team played every other team.
(c) The teams play in distinct leagues.
13. Suppose the edges of a graph represent streets
14. If the vertices of a graph represent cities and the
that must be plowed after a snowstorm. To eulerize the
edges of a graph represent flight routes for a
graph, four edges must be added. The real-world
particular airline, then which of the objects below best
interpretation of this is that
models a pilot’s daily schedule?
(a) four streets will not be plowed.
(a) A path
(b) four streets will be traversed twice.
(b) An Euler circuit
(c) four new streets would be built.
(c) A graph
15. In the graph below, add one or more edges to
produce a graph that has an Euler circuit.
CHAPTER 2
1. Which of the following describes a Hamiltonian
2. Using the nearest-neighbor algorithm and starting at
circuit for the graph below?
vertex A, find the cost of the Hamiltonian circuit for
the graph below.
(a) ABCDEA
(b) ABEDCBDAC
(c) ACDEBA
(a) 25
(b) 27
(c) 26
3. The nearest-neighbor traveling salesman tour for the
4. Using the sorted-edges algorithm, find the cost of
following graph starting at B is
the Hamiltonian circuit for the graph below.
(a) BCDAB
(b) BDCAB
(c) BCADB
5. Suppose that after a hurricane, a van is dispatched
(a) 25
(b) 26
(c) Another answer
6. The graph below has
to pick up five nurses at their homes and bring them
back to work at the local hospital. Which of these
techniques is most likely to be useful in solving this
problem?
(a) no Hamiltonian circuit and no Euler circuit.
(a) Finding an Euler circuit in a graph
(b) an Euler circuit and a Hamiltonian circuit.
(b) Solving a TSP (traveling salesman problem)
(c) no Hamiltonian circuit, but it has an Euler circuit.
(c) Finding a minimum-cost spanning tree in a graph
7. Apply the nearest-neighbor method (starting at
vertex A) to find a cheap tour.
8. Apply the sorted-edges method to find a cheap tour.
CHAPTER 3
1. Given the order-requirement digraph below (time in
2. A vertex coloring seeks to color the vertices of a
minutes) and the priority list T1, T2, T3, T4, T5, T6, apply
graph to ensure which of the following traits?
the list-processing algorithm to construct a schedule
using two processors. How much time does the resulting
schedule require?
(a) Every color is used.
(b) Every edge connects vertices of the same color.
(c) Vertices of the same color are never connected by
(a) 11 minutes
(b) 13 minutes
(c) 14 minutes
an edge.
3. The minimum number of colors needed to color the
4. Determine the minimum number of colors, and how
vertices of the accompanying graph is
often each color is used, in a vertex coloring of the
graphs below:
(a) 4
(b) 2
(c) 3
T1, T2, T3, T4, T5, T6