1. Lecture 19: CONNECTIVITY Sections 8.1 - 8.3
CS1050: Understanding and Constructing Proofs
Spring 2006
Lecture 19: CONNECTIVITY Sections
This work is carried by Jarek Rossignac and one of his PhD students, Jason Williams
Jarek Rossignac

2. Lecture Objectives
Learn graph terminology

3. What are the types of graph?
Graph G(V,E)
V = set of vertices (non-empty)
E = set of edges (unordered pairs of distinct elements of V)
Loop
Multiple edge
Simple Graph
Multigraph
Pseudograph
Directed graph

4. Graph Types type edges multiple loops Simple graph undirected No
Multigraph
Yes
Pseudograph
Directed graph
directed
Directed multigraph

5. Examples of graphs Simple: Multigraph (multiple edges):
Pseudograph (multiple edges and loops):
Directed (loops):
Directed multigraph (multiple edges and loops):

6. Classify graphs Edge between A and B means: They know each other
A is a parent of B
They compete
A has called B
Page A has a link to page B
Have collaborated
A has beaten B in round-robin

7. What is adjacency and incidence?
In an undirected graph
An edge E between vertices A and B is incident with them.
A and B are the endpoints of E
E connects A and B
Vertices A and B are adjacent (neighbors) when there is an edge incident with both

8. What is the degree of a vertex?
In an undirected graph with e edges:
The degree deg(V), also called valence, of vertex V is the number of times V is used by an edge (twice by an incident loop).
A vertex with degree one is pendent (dead end).
A vertex with degree zero is isolated.
The sum of the degrees of all vertices if 2e.
There is an even number of edges of odd degree.

9. Directed graph terminology
E is a directed edge from A to B (denoted AB)
A is adjacent to B
A is the initial vertex of E
B is adjacent from A
B is the terminal or end vertex of E
A=B if E is a loop
In-degree deg–(V) of vertex V is the number of edges for which it is a terminal vertex
Out-degree deg+(V) of vertex V is the number of edges for which it is an initial vertex

10. Cycles A cycle Cn is has n vertices and n-edge the form a cycle
C3 is a triangle C5

11. Complete graphs Kn
A complete graph Kn of n vertices is a simple graph with one edge between each pair
K3 is a triangle K5

12. Wheels
A wheel Wn is a cycle with n vertices plus an additional vertex connected to all W5

13. Bipartite graphs
A graph is bipartite when itd vertices can be colored (red/green) so that each edge joins vertices of different colors
It is complete bipartite if there is an edge between each pair of vertices of different color

14. Subgraph A subgraph of G has a subset of the edges and vertices of G
It must include all the vertices bounding all its edges!

15. Representing graphs Vertices (x,y) , edges (a,b)
Adjacency list: vertices (x, y, a, b, …)
Adjacency matrix
Simple graphs (binary, symmetric)
Multiple graph: integer entries count number of edges
Loops on diagonal
Incidence matrix: edges/vertices
Two 1s per column

16. Isomorphism
Two graphs G and H are isomorphic if there is a bijection between their vertices that leads to the same set of edges.
Expensive to compute, since there are n! vertex/label assignments
Necessary conditions (invariants) help quickly decide that two graphs are NOT isomorphic
same number of vertices and edges
same degree list

17. Assigned Reading
8.1, 8.2, 8.3

18. Assigned Homework P 544-545: 3, 4, 5, 6, 7 P 555: 12, 27, 29f, 36, 42
P 562: 1, 10, 38, 39, 49, 57a, 68

19. Assigned Project
P9: Spanning tree