Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
Directed Graph and Binary Trees
Math 301
Dr. Nahid Sultana
December 19, 2012
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
Introduction
Directed Graph
Basic Definitions
Degrees
Paths
Adjacency matrix and Linked Representation
Binary Trees
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
Directed graphs are graphs in which the edges are one-way.
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This type of graphs are frequently more useful in various
dynamic systems such as digital computers or flow systems.
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In this lecture the basic definitions and properties of directed
graphs will be discussed.
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Many of the definitions are similar to those in the last chapter
on graph theory.
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The binary trees will also be discussed in this lecture which is
a fundamental structure in mathematics and computer science
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
A directed graph G (V , E ) or digraph (or simply graph)
consists of two things:
1. A set V whose elements are called vertices, nodes, or points.
2. A set E of ordered pairs (u, v ) of vertices called arcs or
directed edges or simply edges.
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We write V (G ) and E (G ) to denote the set of vertices and
the set of edges of a graph G, respectively.
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
Suppose e = (u, v ) is a directed edge in a digraph G .
Then the following terminology is used:
1. e begins at u and ends at v .
2. u is the origin or initial point of e,
and v is the destination or terminal point of e.
3. v is a successor of u.
4. u is adjacent to v , and v is adjacent from u.
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
Suppose e = (u, v ) is a directed edge in a digraph G .
Then the following terminology is used:
1. e begins at u and ends at v .
2. u is the origin or initial point of e,
and v is the destination or terminal point of e.
3. v is a successor of u.
4. u is adjacent to v , and v is adjacent from u.
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If u = v , then e is called a loop.
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
Suppose e = (u, v ) is a directed edge in a digraph G .
Then the following terminology is used:
1. e begins at u and ends at v .
2. u is the origin or initial point of e,
and v is the destination or terminal point of e.
3. v is a successor of u.
4. u is adjacent to v , and v is adjacent from u.
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If u = v , then e is called a loop.
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The directed edges with the same initial point and same
terminal point are said to be parallel.
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
Example: Draw the directed graph consists of:
four vertices: V (G ) = {A, B, C , D} and
seven edges: E (G ) = {e1, e2, ..., e7} =
{(A, D), (B, A), (B, A), (D, B), (B, C ), (D, C ), (B, B)}
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
Example: Draw the directed graph consists of:
four vertices: V (G ) = {A, B, C , D} and
seven edges: E (G ) = {e1, e2, ..., e7} =
{(A, D), (B, A), (B, A), (D, B), (B, C ), (D, C ), (B, B)}
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If the edges and/or vertices of a directed graph labeled with
some kind of data, then the directed graph is called labeled
directed graph.
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Example: In class.
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
I
I
Degrees
Paths
Suppose G is a directed graph.
The outdegree of a vertex v of G , written outdeg(v ), is the
number of edges beginning at v .
And the indegree of v , written indeg(v ), is the number of
edges ending at v .
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
I
I
I
Degrees
Paths
Suppose G is a directed graph.
The outdegree of a vertex v of G , written outdeg(v ), is the
number of edges beginning at v .
And the indegree of v , written indeg(v ), is the number of
edges ending at v .
Since each edge begins and ends at a vertex we have the
following theorem:
Theorem: The sum of the outdegrees of the vertices of a
digraph G equals the sum of the indegrees of the vertices,
which equals the number of edges in G.
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
I
I
I
I
I
Degrees
Paths
Suppose G is a directed graph.
The outdegree of a vertex v of G , written outdeg(v ), is the
number of edges beginning at v .
And the indegree of v , written indeg(v ), is the number of
edges ending at v .
Since each edge begins and ends at a vertex we have the
following theorem:
Theorem: The sum of the outdegrees of the vertices of a
digraph G equals the sum of the indegrees of the vertices,
which equals the number of edges in G.
A vertex v with zero indegree is called a source,
and a vertex v with zero outdegree is called a sink.
Example: In class.
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
Degrees
Paths
The concepts of path, simple path, trail, and cycle in directed
graph are similar as in nondirected graphs except that the
directions of the edges must agree with the direction of the
path.
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
Degrees
Paths
I
The concepts of path, simple path, trail, and cycle in directed
graph are similar as in nondirected graphs except that the
directions of the edges must agree with the direction of the
path.
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A vertex v is reachable from a vertex u if there is a path
from u to v .
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If v is reachable from u, then there must be a simple path
from u to v (just by eliminating redundant edges).
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
In class.
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
Example:
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
I
We will use the term node, rather than vertex, with binary
trees.
A binary tree T is defined as a finite set of elements, called
nodes, such that:
1. T is empty (called the null tree or empty tree), or
2. T contains a distinguished node R, called the root of T , and
the remaining nodes of T form an ordered pair of disjoint
binary trees T1 and T2 .
Math 301
Directed Graph and Binary Trees
Outline
Introduction
Directed Graph
Basic Definitions
Adjacency matrix and Linked Representation
Binary Trees
I
I
We will use the term node, rather than vertex, with binary
trees.
A binary tree T is defined as a finite set of elements, called
nodes, such that:
1. T is empty (called the null tree or empty tree), or
2. T contains a distinguished node R, called the root of T , and
the remaining nodes of T form an ordered pair of disjoint
binary trees T1 and T2 .
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If T does contain a root R, then the two trees T1 and T2 are
called, respectively, the left and right subtrees of R.
If T1 is nonempty, then its root is called the left successor of
R;
If T2 is nonempty, then its root is called the right successor
of R.
A node with no successors is called a terminal node.
Math 301
Directed Graph and Binary Trees