Honors Discrete Chapter 5 Test Review Guide
KEY TERMS. (8 Direct Definitions)
1)
2)
3)
4)
5)
Loop
Multiple Edges
Isolated Vertex
Vertex Set
Edge Set
6) Adjacent
7) Degree of a Vertex
8) Path
9) Circuit
10) Connected Graph
11) Bridge
12) Euler Path
13) Euler Circuit
SOME TYPES OF PROBLEMS TO EXPECT
For the following graph:
1) Label any multiple edges, loops,
isolated vertices, or bridges.
A
B
C
G
2) Find the vertex set, edge set, and
degree of each vertex
D
E
F
For each prompt, DRAW a graph satisfying the characteristics:
1) Vertex Set = {A, B, C, D, E, F} and
Edge Set = {AB, AC, AD, AF, BC, BE, CD, CE, DE, DF, FF}
2) A is adjacent to B and E. B is also adjacent to E. C is also adjacent to D
and C. D is also adjacent to A and E. E is also adjacent to B.
3) The graph has 3 bridges and 4 vertices.
4) The graph has 4 odd vertices and 2 even vertices.
5) Disconnected in which each component has a bridge.
6) Has at least 2 multiple edges, 3 loops, 5 vertices, and 12 total edges.
7) Euler Path and odd vertices are not adjacent.
GRAPH MODEL - draw a graphical representation of the following:
What should be your edges and vertices?
#1: Represent the enclosed neighborhood
of homes and streets below. Shaded
Regions are actual homes.
#2: A river separates 5 islands from its north and
south shore. Bridges now connect the islands to
the shores.
Isle #4
Isle #1
Isle #2
Isle #3
Isle #5
Does the Graph have an Euler Path, Euler Circuit, or Neither?
If the graph has an Euler Path or Euler Circuit then find one.
Find an Eulerization of the graphs:
Find a Semi-Eulerization of the graph:
E
E
S
S
Honors Discrete Chapter 5 Test Review Guide SOLUTIONS
SOME TYPES OF PROBLEMS TO EXPECT
For the following graph:
Multiple Edges: BE, BC
Loops: EE
Bridges: AB, BD, EF, FG
Isolated Vertex: None
A
B
C
G
D
Vertex set: A, B, C, D, E, F, G
Edge set: AB, BC, BC, BC, BE, BE, BD, CE, EE, EF, FG
Degree of each vertex:
A=1 B=7 C=4 D=1
E=6 F=2 G=1
E
F
For each of the following prompts DRAW a graph satisfying the characteristics:
1) Vertex Set = {A, B, C, D, E, F},
4) The graph has 4 odd vertices and 2 even
Edge Set = {AB, AC, AD, AF, BC, BE, CD,
vertices.
CE, DE, DF, FF}
B
C
A
D
5) Disconnected in which each component
has a bridge.
F
E
2) A is adjacent to B and E. B is also
adjacent to E. C is also adjacent to D and
C. D is also adjacent to A and E. E is also
adjacent to B.
C
B
6) Has at least 2 multiple edges, 3 loops, 5
vertices, and 12 total edges.
D
A
E
7) Euler Path and odd vertices are not
adjacent.
3) The graph has 3 bridges and 4 vertices.
Give a graphical representation of the following pictures: What should be edges and vertices?
Land (Island and Shore)
Streets = Edges
=
Vertices
Intersections = Vertices
Bridges = Edges
Does the Graph have an Euler Path, Euler Circuit, or Neither?
If the graph has an Euler Path or Euler Circuit then find one.
1
9
Neither. 4
vertices of
degree 5
8
26
20
1
8
9
7
10 11
18
24
12
13
3
14
4
23
22
21
18 17
6
16
15
5
3
12
19
21 22
13
16
20
4
15 14 23
6
5
17
19
2
11
10
25
7
Euler Circuit.
No vertices of
odd degree
2
Euler Circuit.
No vertices
of odd
degree
Euler Path. 2
vertices of
degree 3
Find an Eulerization of the graphs:
Find a Semi-Eulerization of the graph:
E
E
S
S