Directed Graph and Binary Trees

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Alan Watkins
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1 and Dr. Nahid Sultana December 19, 2012 and

2 Degrees Paths and

3 Directed graphs are graphs in which the edges are one-way. This type of graphs are frequently more useful in various dynamic systems such as digital computers or flow systems. In this lecture the basic definitions and properties of directed graphs will be discussed. Many of the definitions are similar to those in the last chapter on graph theory. The binary trees will also be discussed in this lecture which is a fundamental structure in mathematics and computer science and

4 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. We write V (G) and E(G) to denote the set of vertices and the set of edges of a graph G, respectively. and

5 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. and

6 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. If u = v, then e is called a loop. and

7 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. If u = v, then e is called a loop. The directed edges with the same initial point and same terminal point are said to be parallel. and

8 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)} and

9 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)} 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. Example: In class. and

10 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. and

11 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. and

12 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. and

13 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. and

14 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. A vertex v is reachable from a vertex u if there is a path from u to v. If v is reachable from u, then there must be a simple path from u to v (just by eliminating redundant edges). and

15 In class. and

16 Example: and

17 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 T 1 and T 2. and

18 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 T 1 and T 2. If T does contain a root R, then the two trees T 1 and T 2 are called, respectively, the left and right subtrees of R. If T 1 is nonempty, then its root is called the left successor of R; If T 2 is nonempty, then its root is called the right successor of R. A node with no successors is called a terminal node. and

Lecture 22 Tuesday, April 10

CIS 160 - Spring 2018 (instructor Val Tannen) Lecture 22 Tuesday, April 10 GRAPH THEORY Directed Graphs Directed graphs (a.k.a. digraphs) are an important mathematical modeling tool in Computer Science,