graphs lect4
TRANSCRIPT
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Graph
Nitin Upadhyay
March 01, 2006
Bits-Pilani Goa campus
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Discussion
Counting Path Example
ConnectednessCut Vertices and Cut Edges
General Graphs
Planar Graphs Planar Embedding
Plane Graph Regions
Plane Graph Degree
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Counting Paths and Adjacency Matrices
Let A be the adjacency matrix of graph G.
The number of paths of length kfrom vi to vj
is equal to (Ak)i,j. (The notation (M)i,j denotes
mi,j where [mi,j] = M.)
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Counting Paths Example
How many paths of length 4 are there from 1
to 5 in the graph?
1 2
3
4 5
e1
e2e3
e4
e5e6
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Adjacency Matrix
1 2
3
4 5
e1
e2e3
e4
e5
e6
The number
of paths of length4 from a to dis the
(1, 5)th entry ofA4.
Since
0 1 0 1 0
1 0 1 0 10 1 0 1 1
1 0 1 0 0
0 1 1 0 0
3
9 3 11 1 6
3 15 7 11 8
11 7 15 3 8
1 11 3 9 6
6 8 8 6 8
The (1, 5)th entry ofA4
is 6, indicating that thereare six different paths of
length 4 between 1 and
5.
1:{1-4-1-2-5}
2:{1-2-1-2-5}
3:{1-4-3-2-5}
4:{1-2-5-2-5}
5:{1-2-3-2-5}
6:{1-2-5-3-5}
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Counting Paths Example
How many paths of length 4 are
there from a to d in the right
graph? The adjacency matrix of the
graph is
Hence, the number
of paths of length
4 from a to d is the
(1, 4)th entry ofA4.
Since
There are 8 paths of length
4 from a to d.
a b
cd
=
0110
1001
1001
0110
A
=
8008
0880
0880
8008
4A
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Connectedness
An undirected graph isconnectediff there is a pathbetween every pair ofdistinct vertices in the graph.
E.g., Graph G1 is connected,since for every pair ofdistinct vertices, there is apath between them.
However, G2 is notconnected. E.g., no path inG2 between vertices a and c.
a b c
d fe
a b c
d fe
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Connectedness
Theorem: There is a simple path between
every pair of distinct vertices of a connectedundirected graph.
Connected component: A graph that is notconnected is the union of two or more
connected subgraphs, each pair of which hasno vertex in common. These disjointconnected subgraphs are called theconnected components of the graph.
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Connected Component Example
The graph His the
union of three disjoint
connected subgraph
H1, H2, and H3. These
three subgraphs are the
connectedcomponents
ofH.
a
c
d
f
e
b gh
H1
H2
H3
H
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Cut Vertices and Cut Edges
A cut vertexorcut edge
separates 1 connected
component into 2 ifremoved.
E.g., The cut vertices ofG
are b, c, and e, since
removing one of these
vertices (and its adjacentedges) disconnects the
graph. The cut edges are
[a, b] and [c, e].
a
c
g
b he
d f
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Directed Connectedness
A directed graph is strongly connectediff
there is a directed path from any vertex toany other vertex in the graph.
It is weakly connectediff the underlying
undirectedgraph (i.e., with edge directionsremoved) is connected.
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Directed Connectedness Example
Are the directed graphs G and H strongly
connected or weakly connected?
a b
c
de
a b
c
de
G H
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Directed Connectedness Example
G is strongly connectedsince there is a directed
path between any twovertices. Hence, G is alsoweakly connected. His notstrongly connected since, forinstance, theres nodirected path from a to b. H
is however weaklyconnected since there is apath between any twovertices with all directionsremoved.
ab
c
de
ab
c
de
G H
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General Graphs
Generally graphs drawn on a piece of paper
permit edges to intersect at points other thanvertices.
These points of intersection are called
crossovers.
And the intersecting edges are said to crossover each other.
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Example Crossovers
Here this graph exhibits three crossovers:
{b, e} crosses over {a, d} and {a, c}. {b, d} crosses over {a, c}
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Planar Graphs
A Graph G is said to be planar if it can be
drawn on a plane without any crossovers.
A graph is non planar if there is no way to
convert it into planar.
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Example conversion
Non planar to planar
c c
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Planar embedding
Graphs can be represented such that no two
edges of the graph intersect except possiblyat a vertex to which they both incident.
A drawing of a geometric representation of a
graph on any surface such that no edges
intersect is called embedding. An graph G is planar if there exists a graph
isomorphic to G that is embedded in a plane.
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Plane Graph Regions
A plane graph G can be thought of as dividing
the plane into regions orfaces.
The regions are the connected portions of the
plane remaining after all the curves and pointsof the plane corresponding, respectively, to
edges and vertices of G have been deleted.
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Plane Graph Regions
A plane graph partitions the plane into
regions of G.
There is exactly one region whose area is
infinite, known as exteriororinfinite region.
Every other region is an interior region.
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Plane Graph Regions
The boundary of a region is the subgraph
formed by the vertices and edgesencompassing that region.
If the boundary of the exterior region of a
plane graph is a cycle, that cycle is knownas the maximal cycle of the graph.
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Plane Graph Regions
The degree of a region is the number of
edges in a (closed) walk encloses it. A bridge belongs to the boundary of only
one region, thus it contributes to the size of
the boundary twice.
The sum of the degrees of all the regions in aplane graph is twice the size if the graph.
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Region Example
Graph has six interior regions and one exterior
regions.a
eb c
d
f
g
h
i
j
Interior regions- 1:{a-b-c-a} 3:{a-c-d-e-a} 5:{g-h-j-g} 6:{g-h-i-f-g}
2:{b-c-d-b} 4:{j-h-i-j}
Exterior region- 1:{a-b-d-e-f-i-j-g-f-e-a}
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Degree example
In the graph there exists four interior regions
of degree 3 and two interior regions ofdegree 4.The degree of exterior region is 10.
a
eb c
d
f
g
h
i
j
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Degree example
The sum of the degrees (of the regions) is
30, and the graph has 15 edges.
a
eb c
d
f
g
h
i
j
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Example 1
Find out the interior and exterior regions,
degree of each region and total degree of theplanar graph G.
a b c d
h g f e
Interior regions:
1:{a-b-g-h-a} 2:{b-c-f-g-b}
3:{c-d-e-f-c} degree: {4}
Exterior region
1: {a-b-c-d-e-f-g-h-a}
Degree: {8}
Total degree: 8 + (4 * 3) = 20
Number of edges: 20/2 = 10
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Example 2
Find out the interior and exterior regions,
degree of each region and total degree of theplanar graph G.Interior regions:
1:{b-d-c-b} 2:{e-f-g-e}
degree: {3}
Exterior region
1: {a-b-c-d-e-f-g-e-d-b-a}
Degree: {10}
Total degree: 10 + (2 * 3) = 16c
b
a
d e
f
g
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Questions Questions ?