graphs lect3

8/14/2019 Graphs Lect3

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Graphs

Nitin Upadhyay

February 27, 2006Bits-Pilani Goa campus


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Discussion

What is a Graph?

Applications of Graphs

Categorization

Terminology


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Special Graph Structures

Special cases of undirected graph structures:

Complete graphs Kn

Cycles Cn Wheels Wn

n-Cubes Qn

Bipartite graphs Complete bipartite graphs Km,n


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Graph Representations

Adjacency-matrix representation

Incidence matrix representation

Edge-set representation

Adjacency-set representation

Adjacency List


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Graph Isomorphism

Formal definition: Simple graphs G1=(V1, E1) and G2=(V2, E2) are

isomorphicif there is a function f:V1V2 such that

f is one-to-one .

f is onto, and

a,bV1, a and b are adjacent in G1 ifff(a) and

f(b) are adjacent in G2. fis the renaming function that makes the two

graphs identical.


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Graph Invariants underIsomorphism

Necessarybut not sufficientconditions for

G1=(V1, E1) to be isomorphic to G2=(V2, E2):

|V1|=|V2|, |E1|=|E2|.

The number of vertices with degree n is the same

in both graphs.

For every proper subgraph gof one graph, there

is a proper subgraph of the other graph that isisomorphic to g.


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Are These Isomorphic?

If isomorphic, label the 2nd graph to show the

isomorphism, else identify difference.

ab

c

d

e

* Same # of

vertices

* Same # of

edges* Different

# of verts ofdegree 2!

(1 vs 3)Hence, they are NOT isomorphic!


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Subgraphs

A subgraph of a graph G=(V, E) is a graph

H=(W, F) where WVand FE.

G H


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Spanning Subgraph

A subgraph H= (W, F) of G= (V, E) is called a

spanning subgraph of G iff:

W(H)=V(G)

F(H) E(G)

G H


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Connectivity

In an undirected graph, apath of length n

from u to vis a sequence ofn adjacent edges

going from vertex u(=x0) to vertex v(=xn).

A path is a circuitifu=vand n > 0.

A path pass through the verticesx1,x2,..,xn-1,

ortraverses the edges e1, e2, , en.

A path is simple if it does not contain the

same edge more than once.


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Path Example

a, d, c, f, e is a simplepath of length 4.

d, e, c, a is not a pathsince {e, c} is not an edge.

b, c, f, e, b is a circuit oflength 4 since this pathbegins and ends at b.

Path a, b, e, d, a, b is nota simple path since itcontains the edge {a, b}

twice.

a b c

d fe


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Counting Paths andAdjacency 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

a b

cd

=

8008

0880

0880

8008

4A

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 dis the

(1, 4)th entry ofA4.Since

There are 8 paths of length

4 from a to d.

a b

cd

a b

cd

=

0110

1001

10010110

A

=

8008

0880

0880

8008

4A


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Euler & Hamilton Paths

An Euler circuitin a graph G is a simple

circuit containing every edge ofG.

An Euler path in G is a simple path

containing every edge ofG.

A Hamilton circuitis a circuit that traverses

each vertex in G exactly once.

A Hamilton path is a path that traverses eachvertex in G exactly once.


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Some Useful Theorems

A connected multigraph has an Euler circuit

iff each vertex has even degree.

A connected multigraph has an Euler path

(but not an Euler circuit) iff it has exactly 2vertices of odd degree.

If (but not only if) G is connected, simple, has

n3 vertices, and vdeg(v)n/2, then G hasa Hamilton circuit.s


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Paths in Directed Graphs

Same as in undirected graphs, but the path

must go in the direction of the arrows.


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Graph Unions

The unionG1G2 of two simple graphs

G1=(V1, E1) and G2=(V2,E2) is the simple

graph (V1V2, E1E2).


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Graph Union Example

Find the union of the graphs G1 and G2

shown below.The vertex set of the union G

1G

2is the union of the

two vertex sets, namely {a, b, c, d, e, f}. The edge

set of the union is the union of the two edge set.

a b c

d e

a b c

d f

a b c

d fe

G1 G2 G1 G2


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Questions

Questions ?

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