graphs lect3
TRANSCRIPT
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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 ?