1. CS 367 – Introduction to Data Structures
Graphs – Part I
CS 367 – Introduction to Data Structures

2. Graph Definitions Vertices Edges General graph definition
nodes in the graph
Edges
connection from one node to another
General graph definition
a graph is a collection of vertices and edges with little or no restrictions on which vertices are connected together

3. Graph vs. Trees In theory, a tree is a restricted graph
it is laid out in a hierarchical manner
each node can have only one parent
each node has a set of children from 0 to n
siblings have no direct connection
In a graph, a “child” can connect directly to its “sibling” and even to its “parent”
the quotes are provided because, unlike trees, there is no notion of parent and child

4. Simple and Directed Graphs
A simple graph has a set of vertices connected by a set of edges
no notion of direction from one vertex to another
an edge from vm to vn is considered the same as and edge from vn to vm
A directed graph (or digraph) also has a set of vertices connected by a set of edges
now there is a notion of direction
an edge from vm to vn is NOT the same as and edge from vn to vm

5. Simple and Directed GraphsB C B C G F G D D F
Simple Graph
(B->D) == (D->B)
Directed Graph
(B->D) != (D->B)
(in fact, there is no edge D->B)

6. Multigraph and Psuedograph
A multigraph allows more than one edge between two vertices
an edge must have two different vertices on each end
A psuedograph is exactly like a multigraph except it allows an edge to have the same vertex at both ends
this is called a cycle

7. Multigraph and PsuedographB C B C D D
Multigraph
Psuedograph

8. Weighted Graph A graph where values are associated with each edge
usually considered the “cost” of moving from one vertex to another vertex
any type of graph can be converted into a weighted graph

9. Weighted Graph A 8 5 E B 9 C 7 5 4 2 D F

10. Representing a Graph Several ways to represent a graph
need to store vertices
need a way to represent edges
Two possibilities
adjacency list
adjacency matrix

11. Adjacency List Stores a list of all vertices adjacent to each vertex
can do this in a table or a linked list
Using a table
simple solution is to use a 2-D array
could use an array (or vector) where each element contains vertex and list
Using a linked list
each node contains two pointers
one pointer to the next vertex
one pointer to a list of adjacent vertices

12. Adjacency List – Table A B C D A B A D F E C A D B C D A B C F D E F F

13. Adjacency List – Linked ListB C D B A D F A E C A D B C D A B C F D E F F B E

14. Adjacency Matrix Implemented using a two dimensional square array
each vertex is listed as a row
each vertex is listed as a column
a 1 in any any element (i, j) indicates the two nodes are adjacent to each other

15. Adjacency Matrix A B C D E F A 1 1 1 A E B 1 1 1 B C C 1 1 D 1 1 1 F D1 1 1 A E B 1 1 1 B C C 1 1 D 1 1 1 F D E 1 F 1 1

16. Which Representation?
Which method to use depends on what is being done
processing vertices adjacent to a vertex?
quicker to use an adjacency list
removing edges?
quicker to use an adjacency matrix