Chapter 15
Graph
Theory
© 2008 Pearson Addison-Wesley.
All rights reserved
Chapter 15: Graph Theory
15.1
15.2
15.3
15.4
Basic Concepts
Euler Circuits
Hamilton Circuits and Algorithms
Trees and Minimum Spanning Trees
15-4-2
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Chapter 1
Section 15-4
Trees and Minimum Spanning Trees
15-4-3
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Trees and Minimum Spanning Trees
•
•
•
•
•
Connected Graphs and Trees
Spanning Trees
Minimum Spanning Trees
Kruskal’s Algorithm
Number of Vertices and Edges
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Trees
In the previous two sections, we looked at
graphs with special kinds of circuits, In
this section we examine graphs (called
trees) that have no circuits.
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Connected Graph
A connected graph is one in which there is
at least one path between each pair of
vertices.
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Example: Connected Graph
Below is an example of a connected graph.
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Tree
We call a graph a tree if the graph is connected
and contains no circuits.
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Example: Trees
Each of the graphs below is a tree.
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Example: Trees
Neither of the graphs below is a tree.
Not
connected
Has a circuit
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Unique Path Property of Trees
In a tree there is always exactly one path
from each vertex in the graph to any other
vertex in the graph.
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Spanning Tree
A spanning tree for a graph is a subgraph
that includes every vertex of the original, and
is a tree.
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Example: Finding a Spanning Tree
Find a spanning tree for the graph below.
Solution
We must break two circuits by removing a single
edge from each. One of several possible ways to
do this is shown.
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Minimum Spanning Tree
A spanning tree that has total minimum total
weight is called a minimum spanning tree
for the graph.
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Kruskal’s Algorithm for Finding a
Minimum Spanning Tree
Choose edges for the spanning tree as follows.
Step 1:
First edge: choose any edge with the
minimum weight.
Step 2:
Next edge: choose any edge with
minimum weight from those not yet
selected. (The subgraph can look
disconnected at this stage.)
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Kruskal’s Algorithm for Finding a
Minimum Spanning Tree
Step 3:
Continue to choose edges of minimum
weight from those not yet selected,
except do not select any edge that
creates a circuit in the subgraph.
Step 4:
Repeat step 3 until the subgraph
connects all vertices of the original
graph.
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Example: Kruskal’s Algorithm
Use Kruskal’s algorithm to find a minimum
spanning tree for the graph.
1
E
7
D
A
10
7.5
1.5
B
4.5
F 9.5
3
4
C
8
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Example: Kruskal’s Algorithm
Solution
First, choose ED (the smallest weight).
1
E
7
D
A
10
7.5
1.5
B
4.5
F 9.5
3
4
C
8
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Example: Kruskal’s Algorithm
Solution
Now choose BF (the smallest remaining weight).
1
E
7
D
A
10
7.5
1.5
B
4.5
F 9.5
3
4
C
8
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Example: Kruskal’s Algorithm
Solution
Now CD and then BD.
1
E
7
D
A
10
7.5
1.5
B
4.5
F 9.5
3
4
C
8
15-4-20
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Example: Kruskal’s Algorithm
Solution
Note EF is the smallest remaining, but that would
create a circuit. Choose AE and we are done.
1
E
7
D
A
10
7.5
1.5
B
4.5
F 9.5
3
4
C
8
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Example: Kruskal’s Algorithm
Solution
The total weight of the tree is 16.5.
1
E
7
D
A
10
7.5
1.5
B
4.5
F 9.5
3
4
C
8
15-4-22
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Number of Vertices
If a graph is a tree, then the number of edges
in the graph is one less than the number of
vertices.
A tree with n vertices has n – 1 edges.
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Example: Vertex/Edge Relation
A molecule of a chemical compound contains 54
atoms and it has a tree-like structure. How many
chemical bonds are there in the molecule?
Solution
If the atoms are like the vertices of a tree, the bonds
are like the edges. This tree has 54 vertices, so the
number of bonds (edges) must be 54 – 1 = 53.
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