Discrete Math
Review Ch. 5 – 7
Given the graph shown, answer the following questions.
1) Vertex A is adjacent to
2) Name the bridge(s) of the graph.
3) Name the odd vertices.
Assume you have a graph with vertices L, M, N, P, Q and edges LM, LQ, MP, MQ, NP, and NN.
4) The degree of vertex Q is
5) Name the bridge(s) of the graph.
6) Name the even vertices.
Use the figure below to answer the following questions.
7) Name the graph(s) with an Euler circuit.
8) Identify each graph as connected or disconnected.
Use the figure below to answer the following questions.
9) Tell whether each drawing has an open, closed or no unicursal tracing.
In a certain city there is a river running through the middle. There are 3 islands and 7 bridges as shown in the figure below.
10) In the graph that models this situation, how many vertices and edges would there be?
11) Give the degree of each vertex.
12) IT IS POSSIBLE to take a walk through this town, starting on the North Bank, crossing each bridge once and only once. Give the
ending location.
13) Suppose that there is a crossing charge of $1.00 every time one crosses a bridge. A tourist wants to start on the North Bank, stroll
accross each of the bridges at least once, and return to her hotel on the North Bank at the end of the trip. What is the cheapest possible
cost of such a trip?
Solve the problem.
14) A graph has 6 vertices - 2 vertices of degree 4, 2 vertices of degree 3, and 2 vertices of degree 2. Give number of edges in the
graph.
15) The degree of each vertex in the complete graph with 100 vertices is
16) The sum of the degrees of the vertices in the complete graph with 100 vertices is
17) The number of edges in the complete graph with 100 vertices is
18) The number of distinct Hamilton circuits in the complete graph with 100 vertices is
19) In a complete graph with 45 edges there are
vertices.
20) In a complete graph with 720 distinct Hamilton circuits, there are
vertices.
A garbage truck must pick up garbage at 4 different dump sites (A, B, C, D) as shown in the graph below, starting and ending
at A. The numbers on the edges represent distances (in miles) between locations. The truck driver wants to minimize the total
length of the trip.
21) The nearest-neighbor algorithm applied to the graph yields the following solution:
22) The cheapest-link algorithm applied to the graph yields the following solution:
23) The repetitive nearest-neighbor algorithm applied to the graph yields the following solution:
A traveling salesman’s territory consists of the 5 cities shown on the following mileage chart. The salesman must organize a
round trip that starts and ends at Louisville (his hometown) and will pass through each of the other 4 cities exactly once.
Mileage Chart
24) The cheapest-link algorithm applied to this problem yields the following solution:
25) At an average cost of 25 cents per mile, the trip found using the cheapest-link algorithm that starts at Louisville and passes
through each of the other cities exactly once would cost
Solve the problem.
26) Suppose T is a tree with 41 vertices. State True or False for each statement below.
a) T must be connected.
b) T can have any number of bridges.
c) T could have any number of loops.
d) T has exactly 40 bridges.
e) There can be multiple ways to get from one vertex to another.
27) Suppose G is a graph with 51 vertices and 50 edges. State True or False for each statement below.
a) If G is a network, then G must be a tree.
b) G is either a tree or is not connected.
c) If G is disconnected, then G must have at least one circuit.
d) G cannot have more than one path joining any two vertices.
28) State whether each graph is a network, a tree, or neither.
29) Give the number of spanning trees in the following graph.
30) Give the number of spanning trees in the following graph.
The questions that follow refer to the problem of finding the minimum spanning tree for the weighted graph shown below.
31) Using Kruskal's algorithm, list the edges in the order in which they would be chosen.
32) The minimum spanning tree includes _________ edges.
33) The total weight of the minimum spanning tree is
The questions that follow refer to the problem of finding the minimum spanning tree for the weighted graph shown below.
34) Using Kruskal's algorithm, list the edges in the order in which they would be chosen.
35) The minimum spanning tree includes _________ edges.
36) The total weight of the minimum spanning tree is
Use the mileage chart shown below to find the minimum spanning tree for the 6 cities shown below.
Mileage Chart
37) Using Kruskal's algorithm, list the edges in the order in which they would be chosen.
38) The minimum spanning tree includes _________ edges.
39) The total weight of the minimum spanning tree is
ANSWER KEY
1) C, D
2) BD
3) B, C, D, E
4) 2
5) MP, NP
6) L, P, Q
7) None
8) Connected: 1 & 4; Disconnected: 2 & 3
9) 1: Open, 2: None, 3: Closed
10) 5 vertices, 7 edges
11) A=2, B=4, C=3, NB=3, SB=2
12) Island C
13) $8
14) 2*4 + 2*3 + 2*2 = 18 degrees = 9 Edges
15) 99
16) 100*99 = 9900
17) 9900 / 2 = 4950
18) 99!
19) 45 edges = 90 degrees total = 9 * 10, therefore 10 Vertices
20) 720 = 6!, therefore 7 Vertices
21) ABDCA (weight = 30)
22) ADBCA or ACBDA
23) ACBDA or ADBCA
24) Louisville-Chicago-Boston-Buffalo-Columbus-Louisville or
Louisville-Columbus-Buffalo-Boston-Chicago-Louisville
25) 2236 * $0.25 = $559
26) a=True, b=False, c=False, d=True, e=False
27) a=True, b=True, c=True, d=False
28) 1: Tree (and Network), 2: Network, 3: Tree (and Network),
4: Neither
29) 5
30) 6*4 = 24
31) CD, BC, AC
32) 4
33) 24
34) BD, BF, EF, AC, CD
35) 5
36) 20.7
37) Dallas-Houston, Atlanta-Memphis, Kansas City-Memphis,
Memphis-Dallas, Denver-Kansas City
38) 5
39) 2117 miles