Lecture 5.4: Paths and Connectivity
CS 250, Discrete Structures, Fall 2011
Nitesh Saxena
*Adopted from previous lectures by Zeph Grunschlag
Course Admin -- Homework 5
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Due at 11am on Nov 30 (Wed)
Covers the chapter on Graphs (lecture 5.*)
Thanksgiving is in the way – please try to
start early
Has a 10-pointer bonus
problem too
Little work
will not hurt!
Lecture 5.4 -- Paths and
Connectivity
Course Admin -- Homework 4
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Grading now
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Expect results in a couple of days
Graded HWs will be distributed next Tuesday
Solution posted
Lecture 5.4 -- Paths and
Connectivity
Course Admin -- Final Exam
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Thursday, December 8, 10:45am1:15pm, lecture room
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Heads up!
Please mark the date/time/place
Emphasis on post mid-term 2 material
Coverage:
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65% post mid-term 2 (lectures 4.*, 5.*, 6.*), and
35% pre mid-term 2 (lecture 1.*. 2.* and 3.*)
Our last lecture will be on December 6
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We plan to do a final exam review then
Lecture 5.4 -- Paths and
Connectivity
Outline
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Paths and Isomorphism
Connectivity
Lecture 5.4 -- Paths and
Connectivity
A visualization of the Linkein Social
Graph
Lecture 5.4 -- Paths and
Connectivity
and of the Facebook network
Lecture 5.4 -- Paths and
Connectivity
Recap: Graph Isomorphism
DEF: Suppose G1 = (V1, E1 ) and G2 = (V2, E2 ) are
pseudographs. Let f :V1V2 be a function
s.t.:
1)
f is bijective
2)
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; or e(u, v, G1) = e(f(u), f(v), G2)
Then f is called an isomorphism and G1 is said to
be isomorphic to G2.
Lecture 5.4 -- Paths and
Connectivity
Properties of Isomorphisms
Two graphs are isomorphic to each other if
they satisfy the following properties:
 same number of vertices
 same number of edges
 same degrees at corresponding vertices
 Any subgraph of one is isomorphic to some
subgraph of the other
Lecture 5.4 -- Paths and
Connectivity
Warm-up Exercise
Theorem: Isomorphism is an equivalence relation
Proof: We need to prove that the isomorphism
relation is reflexive, symmetric, and transitive.
Let’s use the whiteboard.
Lecture 5.4 -- Paths and
Connectivity
Paths
A path in a graph is a continuous way of getting
from one vertex to another by using a
sequence of edges.
1
e3
e1
e2
3
e6
2
e4
e5
e7
4
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
Paths
A path in a graph is a continuous way of getting
from one vertex to another by using a
sequence of edges.
1
e3
e1
e2
3
e6
2
e4
e5
e7
4
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
Paths
A path in a graph is a continuous way of getting
from one vertex to another by using a
sequence of edges.
1
e3
e1
e2
3
e6
2
e4
e5
e7
4
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
Paths
A path in a graph is a continuous way of getting
from one vertex to another by using a
sequence of edges.
1
e3
e1
e2
3
e6
2
e4
e5
e7
4
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
Paths
A path in a graph is a continuous way of getting
from one vertex to another by using a
sequence of edges.
1
e3
e1
e2
3
e6
2
e4
e5
e7
4
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
Paths
A path in a graph is a continuous way of getting
from one vertex to another by using a
sequence of edges.
1
e3
e1
e2
3
e6
2
e4
e5
e7
4
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
Paths
A path in a graph is a continuous way of getting
from one vertex to another by using a
sequence of edges.
1
e3
e1
e2
3
e6
2
e4
e5
e7
4
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
Paths
A path in a graph is a continuous way of getting
from one vertex to another by using a
sequence of edges.
1
e3
e1
e2
3
e6
2
e4
e5
e7
4
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
Paths in Real (CS) World
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Linkedin paths (let us see in practice)
Facebook paths
Internet paths
…
Lecture 5.4 -- Paths and
Connectivity
Paths and Isomorphism
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Paths and circuits can also be serve as a
good criteria for determining whether two
graphs are isomorphic
Look for (simple) circuits of different
lengths. Example below:
u1
u2
u6
u3
u5
u4
Lecture 5.4 -- Paths and
Connectivity
Number of Paths of Certain Lengths
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a
d
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
b
c
0

1
A
1

0

1
1
0
0
0
0
1
1
Lecture 5.4 -- Paths and
Connectivity
0

1
1

0 
Number of Paths of Certain Lengths
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
a
d
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
b
c
0

1
A2  
1

0

1
1
0
0
0
0
1
1
0  0

1  1
1  1

0  0
1
1
0
0
0
0
1
1
Lecture 5.4 -- Paths and
Connectivity
0  2
 
1 0


1
0
 
0   2
0
0
2
2
2
2
0
0
2

0
0

2 
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.
Lecture 5.4 -- Paths and
Connectivity
Connectivity
Q: Which of the following graphs are
connected?
1
2
3
4
Lecture 5.4 -- Paths and
Connectivity
Connectivity
A: First and second are disconnected. Last is
connected.
1
2
3
4
Lecture 5.4 -- Paths and
Connectivity
Connectivity
A: First and second are disconnected. Last is
connected.
1
2
3
4
Lecture 5.4 -- Paths and
Connectivity
Connectivity
A: First and second are disconnected. Last is
connected.
1
2
3
4
Lecture 5.4 -- Paths and
Connectivity
Connectivity
A: First and second are disconnected. Last is
connected.
1
2
3
4
Lecture 5.4 -- Paths and
Connectivity
Connectivity
A: First and second are disconnected. Last is
connected.
1
2
3
4
Lecture 5.4 -- Paths and
Connectivity
Connectivity
A: First and second are disconnected. Last is
connected.
1
2
3
4
Lecture 5.4 -- Paths and
Connectivity
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:
rob
job
jab
Q: Is “fun” connected to “car” ?
Lecture 5.4 -- Paths and
Connectivity
English Connectivity Puzzle
A: Yes: funfanfarcar
Or: funfinbinbanbarcar
Lecture 5.4 -- Paths and
Connectivity
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
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
Some Theorems
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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.
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
1
the following graph?
6
5
Lecture 5.4 -- Paths and
Connectivity
2
7
8
3
4
Connected Components
A: The components are {1,3,5},{2,4,6},{7} and
{8} as one can see visually by pulling
components apart:
1
6
5
2
8
3
4
Lecture 5.4 -- Paths and
Connectivity
7
Connected Components
A: The components are {1,3,5},{2,4,6},{7} and
{8} as one can see visually by pulling
components apart:
1
6
2
5
3
4
Lecture 5.4 -- Paths and
Connectivity
7
8
Connected Components
A: The components are {1,3,5}, {2,4,6}, {7}
and {8} as one can see visually by pulling
components apart:
6
1
2
7
5
3
4
Lecture 5.4 -- Paths and
Connectivity
8
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)
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
Lecture 5.4 -- Paths and
Connectivity
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.
Lecture 5.4 -- Paths and
Connectivity
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.
Lecture 5.4 -- Paths and
Connectivity
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 nonobvious:
1)
Should we ignore directions?
2)
Should we insist that we can get from a
to b in actual digraph?
3)
Should we insist that we can get from a
to b and that we can get from b to a?
Lecture 5.4 -- Paths and
Connectivity
Connectivity in
Directed Graphs
Resolution: Don’t bother choosing which definition is
better. Just define to separate concepts:
1)
Weakly connected : can get from a to b in
underlying undirected graph
2)
Semi-connected (my terminology): can get from
a to b OR from b to a in digraph
3)
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.
Lecture 5.4 -- Paths and
Connectivity
Connectivity in
Directed Graphs
Q: Classify the connectivity of each graph.
Lecture 5.4 -- Paths and
Connectivity
Connectivity in
Directed Graphs
A:
semi
weak
Lecture 5.4 -- Paths and
Connectivity
strong
Today’s Reading
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Rosen 10.3 and 10.4
Little work
will not hurt!
Lecture 5.4 -- Paths and
Connectivity