1. Lecture 5.4: Paths and Connectivity
CS 250, Discrete Structures, Fall 2011
Nitesh Saxena
*Adopted from previous lectures by Zeph Grunschlag

2. Course Admin -- Homework 4
Graded and returned now
Please pick them up if you haven’t done so already
Solution posted
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Lecture Paths and Connectivity

3. Course Admin -- Homework 5
Due at 11am on Dec 4 (Tues)
Covers the chapter on Graphs (lecture 5.*)
Has a 25-pointer bonus
problem too
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Lecture Paths and Connectivity

4. Course Admin -- Final Exam
Tuesday, December 11,  10:45am- 1:15pm, lecture room
Please mark the date/time/place
Emphasis on post mid-term 2 material
Coverage:
65% post mid-term 2 (lectures 4.*, 5.*), and
35% pre mid-term 2 (lecture 1.*. 2.* and 3.*)
Our last lecture will be on December 4
We plan to do a final exam review then
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Lecture Paths and Connectivity

5. Lecture 5.4 -- Paths and Connectivity
Outline
Paths
Connectivity
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Lecture Paths and Connectivity

6. Paths
A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges.
EG: could get from 1 to 3 circuitously as follows:
1-e12-e11-e33-e42-e62-e52-e43 e6 e1 e2 1 2 e5 e3 e4 e7 3 4
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Lecture Paths and Connectivity

7. Paths
A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges.
EG: could get from 1 to 3 circuitously as follows:
1-e12-e11-e33-e42-e62-e52-e43 e6 e1 e2 1 2 e5 e3 e4 e7 3 4
4/12/2018
Lecture Paths and Connectivity

8. Paths
A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges.
EG: could get from 1 to 3 circuitously as follows:
1-e12-e11-e33-e42-e62-e52-e43 e6 e1 e2 1 2 e5 e3 e4 e7 3 4
4/12/2018
Lecture Paths and Connectivity

9. Paths
A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges.
EG: could get from 1 to 3 circuitously as follows:
1-e12-e11-e33-e42-e62-e52-e43 e6 e1 e2 1 2 e5 e3 e4 e7 3 4
4/12/2018
Lecture Paths and Connectivity

10. Paths
A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges.
EG: could get from 1 to 3 circuitously as follows:
1-e12-e11-e33-e42-e62-e52-e43 e6 e1 e2 1 2 e5 e3 e4 e7 3 4
4/12/2018
Lecture Paths and Connectivity

11. Paths
A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges.
EG: could get from 1 to 3 circuitously as follows:
1-e12-e11-e33-e42-e62-e52-e43 e6 e1 e2 1 2 e5 e3 e4 e7 3 4
4/12/2018
Lecture Paths and Connectivity

12. Paths
A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges.
EG: could get from 1 to 3 circuitously as follows:
1-e12-e11-e33-e42-e62-e52-e43 e6 e1 e2 1 2 e5 e3 e4 e7 3 4
4/12/2018
Lecture Paths and Connectivity

13. Paths
A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges.
EG: could get from 1 to 3 circuitously as follows:
1-e12-e11-e33-e42-e62-e52-e43 e6 e1 e2 1 2 e5 e3 e4 e7 3 4
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Lecture Paths and Connectivity

14. Paths in Real (CS) World
Linkedin paths
Facebook paths
Internet paths …
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Lecture Paths and Connectivity

15. Number of Paths of Certain Lengths
Adjacency matrix A for a graph depicts all paths of length 1
The matrix A2 depicts number of paths of length 2
In general, the matrix Ak depicts number of paths of length k a b c d
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Lecture Paths and Connectivity

16. Number of Paths of Certain Lengths
Adjacency matrix A for a graph depicts all paths of length 1
The matrix A2 depicts number of paths of length 2
In general, the matrix Ak depicts number of paths of length k a b c d
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Lecture Paths and Connectivity

17. Lecture 5.4 -- Paths and Connectivity
DEF: Let G be a pseudograph. Let u and v be vertices. u and v are connected to each other if there is a path in G which starts at u and ends at v. G is said to be connected if all vertices are connected to each other.
Note: Any vertex is automatically connected to itself via the empty path.
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Lecture Paths and Connectivity

18. Lecture 5.4 -- Paths and Connectivity
Q: Which of the following graphs are connected? 1 2 3 4
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Lecture Paths and Connectivity

19. Lecture 5.4 -- Paths and Connectivity
A: First and second are disconnected. Last is connected. 1 2 3 4
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Lecture Paths and Connectivity

20. Lecture 5.4 -- Paths and Connectivity
A: First and second are disconnected. Last is connected. 1 2 3 4
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Lecture Paths and Connectivity

21. Lecture 5.4 -- Paths and Connectivity
A: First and second are disconnected. Last is connected. 1 2 3 4
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Lecture Paths and Connectivity

22. Lecture 5.4 -- Paths and Connectivity
A: First and second are disconnected. Last is connected. 1 2 3 4
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Lecture Paths and Connectivity

23. Lecture 5.4 -- Paths and Connectivity
A: First and second are disconnected. Last is connected. 1 2 3 4
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Lecture Paths and Connectivity

24. Lecture 5.4 -- Paths and Connectivity
A: First and second are disconnected. Last is connected. 1 2 3 4
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Lecture Paths and Connectivity

25. English Connectivity Puzzle
Can define a puzzling graph G as follows:
V = {3-letter English words}
E : two words are connected if we can get one word from the other by changing a single letter.
One small subgraph of G is:
Q: Is “fun” connected to “car” ?
rob
job
jab
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Lecture Paths and Connectivity

26. English Connectivity Puzzle
A: Yes: funfanfarcar
Or: funfinbinbanbarcar
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Lecture Paths and Connectivity

27. Some Little Theorems
Thm1: Every connected graph with n vertices has at least n-1 edges
Proof: Use proof by mathematical induction
Basis Step (n=1): No. of edges is 0, which is less >= 0 (n-1)
Induction Step: Assume to be true for n = k and show it to be true for n = k + 1
We assumed that a connected graph with k vertices will have at least k–1 edges. Now, we add a new vertex to this graph to obtain a new graph with k vertices.
For the new graph to remain connected, the new vertex should be incident with at least one edge which is also incident with one of the vertices in the old graph. This means that the new graph should have a total of at least (k – 1) + 1 = k edges.
This proves the induction step.
Finally, combining the basis and induction steps, we get that the theorem is true for all n

28. Some Theorems
Thm 2: Vertex connectedness in a simple graph is an equivalence relation
Proof: We will show that the “connectedness” relation is reflexive, symmetric and transitive
It is reflexive since every vertex is connected to itself via a path of length 0
It is symmetric because if a vertex u is connected to another vertex v, then there exists a path between u and v – just traverse the reverse path
It is transitive because if u and v are connected (via path p) and v and w are connected (via path q), then u and w are connected via a path p|q
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29. Lecture 5.4 -- Paths and Connectivity
Some Theorems
Thm 3: If a connected simple graph G is the union of graphs G1 and G2, then G1 and G2 must have a common vertex
Proof: (very simple) let’s use the board.
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Lecture Paths and Connectivity

30. Lecture 5.4 -- Paths and Connectivity
Connected Components
DEF: A connected component (or just component) in a graph G is a set of vertices such that all vertices in the set are connected to each other and every possible connected vertex is included.
Q: What are the connected components of the following graph? 1 6 2 7 8 5 3 4
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Lecture Paths and Connectivity

31. Lecture 5.4 -- Paths and Connectivity
Connected Components
A: The components are {1,3,5},{2,4,6},{7} and {8} as one can see visually by pulling components apart: 3 5 1 6 2 8 7 4
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Lecture Paths and Connectivity

32. Lecture 5.4 -- Paths and Connectivity
Connected Components
A: The components are {1,3,5},{2,4,6},{7} and {8} as one can see visually by pulling components apart: 3 5 1 6 2 7 8 4
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Lecture Paths and Connectivity

33. Lecture 5.4 -- Paths and Connectivity
Connected Components
A: The components are {1,3,5}, {2,4,6}, {7} and {8} as one can see visually by pulling components apart: 3 5 1 6 2 7 8 4
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Lecture Paths and Connectivity

34. Degree of Connectivity
Not all connected graphs are treated equal!
Q: Rate following graphs in terms of their design value for computer networks: 1) 2) 3) 4)

35. Degree of Connectivity
A: Want all computers to be connected, even if 1 computer goes down:
1) 2nd best. However, there
is a weak link— “cut vertex”
2) 3rd best. Connected
but any computer can disconnect
3) Worst!
Already disconnected
4) Best! Network dies
only with 2 bad computers
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Lecture Paths and Connectivity

36. Degree of Connectivity
The network is best because it can only become disconnected when 2 vertices are removed. In other words, it is 2-connected. Formally:
DEF: A connected simple graph with 3 or more vertices is 2-connected if it remains connected when any vertex is removed. When the graph is not 2-connected, we call the disconnecting vertex a cut vertex.
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Lecture Paths and Connectivity

37. Degree of Connectivity
There is also a notion of N-Connectivity where we require at least N vertices to be removed to disconnect the graph.
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Lecture Paths and Connectivity

38. Connectivity in Directed Graphs
In directed graphs may be able to find a path from a to b but not from b to a. However, Connectivity was a symmetric concept for undirected graphs. So how to define directed Connectivity is non-obvious:
Should we ignore directions?
Should we insist that we can get from a to b in actual digraph?
Should we insist that we can get from a to b and that we can get from b to a?
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Lecture Paths and Connectivity

39. Connectivity in Directed Graphs
Resolution: Don’t bother choosing which definition is better. Just define to separate concepts:
Weakly connected : can get from a to b in underlying undirected graph
Semi-connected (my terminology): can get from a to b OR from b to a in digraph
Strongly connected : can get from a to b AND from b to a in the digraph
DEF: A graph is strongly (resp. semi, resp. weakly) connected if every pair of vertices is connected in the same sense.
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Lecture Paths and Connectivity

40. Connectivity in Directed Graphs
Q: Classify the connectivity of each graph.
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41. Connectivity in Directed Graphs
semi weak strong
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42. Lecture 5.4 -- Paths and Connectivity
Today’s Reading
Rosen 10.3 and 10.4
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Lecture Paths and Connectivity