1. Mathematics for Computer Science MIT 6.042J/18.062J
Graphs
Copyright © Radhika Nagpal, 2002.
Prof. Albert Meyer & Dr. Radhika Nagpal

2. Connectivity
Are two vertices connected? Is there a path from one vertex to another.
How many paths are there from u to v …

3. Example: MIT Buildings
Already saw one example last week
Can we get from one building to another without going outside?
= is the graph one connected component?
= is the transitive closure =? VxV

4. Smallest Connected Graph
Question: repeat last part of MIT connected buildings question.
Ask how many people did this question? Answer: n-1edges, now we can prove it

5. Smallest Connected Graph
What does this Graph look like?
One answer: line
Actually any tree !
Lots of choices, could easily optimize for something, e.g. congestion vs shortest distance between nodes (e.g star formation)

6. Trees
Definitions:

7. In class exercise: isomorphism and connectedness false proof.

8. Cut Edge
Not only concerned with whether soemthing is connected but also how easy is it to disconnect it? Fault-tolerance
Definition: An edge is a cut edge if removing it from the graph

9. Cut-edge In the following graph are A or B cut edges?
Anything interesting about A or B that distinguishes them?
An edge is a cut edge iff it is not part of a cycle.

10. Discussion of False Proof
Counter Example
Problem with False Proof 1
Problem with False Proof 2

11. Discussion of cut-edge
Use this theorem on particular graphs
In a tree, every edge is a cut edge
In a mesh, no edge is a cut edge

12. Old slides 1

13. MIT Building Connections26 13 12 10 4 8

14. MIT Building Connections4 13 10 12 26 8 R 4 13 10 12 26 8 26 13 12 10 4 8

15. Composition using Matrices =

16. Composition and Path Lengths
is the set of all pairs (a,b) such that bldg a and b are connected via < 1 other bldg
In general is the set of all pairs (a,b) such that bldg a and b are connected via < k-1 other bldgs.
Prove using induction
Note: different if edges are not bidirectional