1. Lecture 5.3: Graph Isomorphism and Paths
CS 250, Discrete Structures, Fall 2012
Nitesh Saxena
Adopted from previous lectures by Zeph Grunschlag

2. Lecture 5.3 -- Graph Isomorphism and Connectivity
Course Admin -- HW4
Grading now – expect results in about a week
Solution has been posted
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Lecture Graph Isomorphism and Connectivity

3. Course Admin -- Homework 5
Heads up: will be posted today
Due on Dec 4
Covers the chapter on Graphs (lecture 5.*)
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Lecture Graph Isomorphism 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 Graph Isomorphism and Connectivity

5. Lecture 5.3 -- Graph Isomorphism and Connectivity
Lecture next Tuesday
I am traveling (NSF in DC) and won’t be here to give the lecture
The TA, Amit Dutta, will cover for me
He is going to talk about homework assignment mostly
Please do attend the lecture
Please cooperate with him
This is the first time he will be lecturing a big class
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Lecture Graph Isomorphism and Connectivity

6. Lecture 5.3 -- Graph Isomorphism and Connectivity
Outline
Graph Isomorphism
Paths
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Lecture Graph Isomorphism and Connectivity

7. Lecture 5.3 -- Graph Isomorphism and Connectivity
Warm-up Exercise
Theorem: No. of edges in an n-cube is n2n-1 Proof: Use mathematical induction. Let’s use the whiteboard.
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Lecture Graph Isomorphism and Connectivity

8. Lecture 5.3 -- Graph Isomorphism and Connectivity
Various mathematical notions come with their own concept of equivalence, as opposed to equality:
Equivalence for sets is bijectivity:
EG {23, 12, 43}  {12, 23, 43}
Equivalence for graphs is isomorphism: EG 
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Lecture Graph Isomorphism and Connectivity

9. Lecture 5.3 -- Graph Isomorphism and Connectivity
Intuitively, two graphs are isomorphic if can bend, stretch and reposition vertices of the first graph, until the second graph is formed. Etymologically, isomorphic means “same shape”.
EG: Can twist or relabel:
to obtain:
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Lecture Graph Isomorphism and Connectivity

10. Graph Isomorphism Undirected Graphs
DEF: Suppose G1 = (V1, E1 ) and G2 = (V2, E2 ) are pseudographs. Let f :V1V2 be a function s.t.:
f is bijective
for all vertices u,v in V1, the number of edges between u and v in G1 is the same as the number of edges between f (u) and f (v ) in G2.
Then f is called an isomorphism and G1 is said to be isomorphic to G2.
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Lecture Graph Isomorphism and Connectivity

11. Graph Isomorphism Digraphs
DEF: Suppose G1 = (V1, E1 ) and G2 = (V2, E2 ) are directed multigraphs. Let f :V1V2 be a function s.t.:
f is bijective
for all vertices u,v in V1, the number of edges from u to v in G1 is the same as the number of edges between f (u) and f (v ) in G2.
Then f is called an isomorphism and G1 is said to be isomorphic to G2.
Note: Only difference between two definitions is the italicized “from” in no. 2 (was “between”).
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Lecture Graph Isomorphism and Connectivity

12. Graph Isomorphism -Example
EG: Prove that
is isomorphic to
First label the vertices: 2 2 1 3 1 3 5 4 5 4
Lecture Graph Isomorphism and Connectivity
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13. Graph Isomorphism -Example
Next, set f (1) = 1 and try to walk around clockwise on the star. 2 2 1 1 3 3 5 4 5 4
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Lecture Graph Isomorphism and Connectivity

14. Graph Isomorphism -Example
Next, set f (1) = 1 and try to walk around clockwise on the star. The next vertex seen is 3, not 2 so set f (2) = 3. 2 2 1 1 3 3 5 4 5 4
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Lecture Graph Isomorphism and Connectivity

15. Graph Isomorphism -Example
Next, set f (1) = 1 and try to walk around clockwise on the star. The next vertex seen is 3, not 2 so set f (2) = 3. Next vertex is 5 so set f (3) = 5. 2 2 1 3 1 3 5 4 5 4
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Lecture Graph Isomorphism and Connectivity

16. Graph Isomorphism -Example
Next, set f (1) = 1 and try to walk around clockwise on the star. The next vertex seen is 3, not 2 so set f (2) = 3. Next vertex is 5 so set f (3) = 5. In this fashion we get f (4) = 2 2 2 1 3 1 3 5 4 5 4
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Lecture Graph Isomorphism and Connectivity

17. Graph Isomorphism -Example
Next, set f (1) = 1 and try to walk around clockwise on the star. The next vertex seen is 3, not 2 so set f (2) = 3. Next vertex is 5 so set f (3) = 5. In this fashion we get f (4) = 2, f (5) = 4. 2 2 1 3 1 3 5 4 5 4
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Lecture Graph Isomorphism and Connectivity

18. Graph Isomorphism -Example
Next, set f (1) = 1 and try to walk around clockwise on the star. The next vertex seen is 3, not 2 so set f (2) = 3. Next vertex is 5 so set f (3) = 5. In this fashion we get f (4) = 2, f (5) = 4. If we would continue, we would get back to f (1) =1 so this process is well defined and f is a isomorphism. 2 2 1 1 3 3 5 4 5 4
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Lecture Graph Isomorphism and Connectivity

19. Graph Isomorphism -Example
Next, set f (1) = 1 and try to walk around clockwise on the star. The next vertex seen is 3, not 2 so set f (2) = 3. Next vertex is 5 so set f (3) = 5. In this fashion we get f (4) = 2, f (5) = 4. If we would continue, we would get back to f (1) =1 so this process is well defined and f is a morphism. Finally since f is bijective, f is an isomorphism. 2 2 1 3 1 3 5 4 5 4
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Lecture Graph Isomorphism and Connectivity

20. Properties of Isomorphims
Since graphs are completely defined by their vertex sets and the number of edges between each pair, isomorphic graphs must have the same intrinsic properties. I.e. isomorphic graphs have the same…
number of vertices and edges
degrees at corresponding vertices
types of possible subgraphs
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Lecture Graph Isomorphism and Connectivity

21. Lecture 5.3 -- Graph Isomorphism and Connectivity
Exercise
Is “isomorphism” on graphs an equivalence relation?
If so, what are the equivalence classes?
Let’s use the whiteboard!
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Lecture Graph Isomorphism and Connectivity

22. Graph Isomorphism -Negative Examples
To show that two graphs are non-isomorphic need to show that no function can exist that satisfies defining properties of isomorphism. In practice, you try to find some intrinsic property that differs between the 2 graphs in question.
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Lecture Graph Isomorphism and Connectivity

23. Graph Isomorphism -Negative Examples
Q: Why are the following non-isomorphic? u2 v2 u3 u1 v1 v3 u4 v4 u5
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Lecture Graph Isomorphism and Connectivity

24. Graph Isomorphism -Negative Examples
A: 1st graph has more vertices than 2nd.
Q: Why are the following non-isomorphic? u2 v2 u3 v3 u1 v1 u4 v4 u5 v5
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Lecture Graph Isomorphism and Connectivity

25. Graph Isomorphism -Negative Examples
A: 1st graph has more edges than 2nd.
Q: Why are the following non-isomorphic? u2 v2 u3 v3 u1 v1 u4 v4 u5 v5
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Lecture Graph Isomorphism and Connectivity

26. Graph Isomorphism -Negative Examples
A: 2nd graph has vertex of degree 1, 1st graph doesn't.
Q: Why are the following non-isomorphic? u1 u2 u3 u4 u5 u6 v1 v2 v3 v4 v5 v6 u7 u8 u9 v7 v8 v9
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Lecture Graph Isomorphism and Connectivity

27. Graph Isomorphism -Negative Examples
A: 1st graph has 2 degree 1 vertices, 4 degree 2 vertex and 2 degree 3 vertices. 2nd graph has 3 degree 1 vertices, 3 degree 2 vertex and 3 degree 3 vertices.
Q: Why are the following non-isomorphic? u1 u2 u3 u4 u5 u6 v1 v2 v3 v4 v5 v6 u7 u8 v7 v8
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Lecture Graph Isomorphism and Connectivity

28. Graph Isomorphism -Negative Examples
A: None of the previous approaches work as there are the same no. of vertices, edges, and same no. of vertices per degree.
LEMMA: If G and H are isomorphic, then any subgraph of G will be isomorphic to some subgraph of H.
Q: Find a subgraph of 2nd graph which isn’t a subgraph of 1st graph. u1 u2 u3 u4 u5 u6 v1 v2 v3 v4 v5 v6 u7 u8 v7 v8
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Lecture Graph Isomorphism and Connectivity

29. Graph Isomorphism -Negative Examples
A: This subgraph is not a subgraph of the left graph.
Why not? Deg. 3 vertices must map to deg. 3 vertices. Since subgraph and left graph are symmetric, can assume v2 maps to u2. Adjacent deg. 1 vertices to v2 must map to degree 1 vertices, forcing the deg. 2 adjacent vertex v3 to map to u3. This forces the other vertex adjacent to v3, namely v4 to map to u4. But then a deg. 3 vertex has mapped to a deg. 2 vertex. u1 u2 u3 u4 u5 u6 v1 v2 v3 v4 v5 v6 u7 u8 v7 v8
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Lecture Graph Isomorphism and Connectivity

30. 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 Graph Isomorphism and Connectivity

31. 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 Graph Isomorphism and Connectivity

32. 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 Graph Isomorphism and Connectivity

33. 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 Graph Isomorphism and Connectivity

34. 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 Graph Isomorphism and Connectivity

35. 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 Graph Isomorphism and Connectivity

36. 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 Graph Isomorphism and Connectivity

37. 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 Graph Isomorphism and Connectivity

38. 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 Graph Isomorphism and Connectivity

39. Lecture 5.3 -- Graph Isomorphism and Connectivity
Paths
DEF: A path of length n in an undirected graph is a sequence of n edges e1, e2, … ,en such that each consecutive pair ei , ei+1 share a common vertex. In a simple graph, one may instead define a path of length n as a sequence of n+1 vertices v0, v1, v2, … ,vn such that each consecutive pair vi , vi+1 are adjacent. Paths of length 0 are also allowed according to this definition.
Q: Why does the second definition work for simple graphs?
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Lecture Graph Isomorphism and Connectivity

40. Lecture 5.3 -- Graph Isomorphism and Connectivity
Paths
A: For simple graphs, any edge is unique between vertices so listing the vertices gives us the edge-sequence as well.
DEF: A simple path contains no duplicate edges (though duplicate vertices are allowed). A cycle (or circuit) is a path which starts and ends at the same vertex.
Note: Simple paths need not be in simple graphs. E.g., may contain loops.
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Lecture Graph Isomorphism and Connectivity

41. Lecture 5.3 -- Graph Isomorphism and Connectivity
Paths
Q: Find a longest possible simple path in the following graph: e6 e1 e2 1 2 e5 e3 e4 e7 3 4
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Lecture Graph Isomorphism and Connectivity

42. Lecture 5.3 -- Graph Isomorphism and Connectivity
Paths
A: The following path from 1 to 2 is a maximal simple path because
simple: each of its edges appears exactly once
maximal: because it contains every edge except the unreachable edge e7
The maximal path: e1,e5,e6,e2,e3,e4 e1 e6 1 e2 2 e5 e3 e4 e7 3 4
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Lecture Graph Isomorphism and Connectivity

43. Paths in Directed Graphs
One can define paths for directed graphs by insisting that the target of each edge in the path is the source of the next edge:
DEF: A path of length n in a directed graph is a sequence of n edges e1, e2, … ,en such that the target of ei is the source ei+1 for each i. In a digraph, one may instead define a path of length n as a sequence of n+1 vertices v0, v1, v2, … ,vn such that for each consecutive pair vi , vi+1 there is an edge from vi to vi+1 .
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Lecture Graph Isomorphism and Connectivity

44. Paths in Directed Graphs
Q: Consider digraph adjacency matrix:
Find a path from 1 to 4.
Is there a path from 4 to 1?
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Lecture Graph Isomorphism and Connectivity

45. Paths in Directed Graphs1
There’s no path from 4 to 1. From 4 must go to 2, from 2 must stay at 2 or return to 4. In other words 2 and 4 are disconnected from 1.
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Lecture Graph Isomorphism and Connectivity

46. Paths in Directed Graphs
132 .
There’s no path from 4 to 1. From 4 must go to 2, from 2 must stay at 2 or return to 4. In other words 2 and 4 are disconnected from 1.
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Lecture Graph Isomorphism and Connectivity

47. Paths in Directed Graphs
1324.
There’s no path from 4 to 1. From 4 must go to 2, from 2 must stay at 2 or return to 4. In other words 2 and 4 are disconnected from 1.
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Lecture Graph Isomorphism and Connectivity

48. Lecture 5.3 -- Graph Isomorphism and Connectivity
Today’s Reading
Rosen 10.3 and 10.4
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Lecture Graph Isomorphism and Connectivity