Graph theory 1

1

GRAPH THEORY
BASIC TERMINOLOGY
PART I

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Basic Graph Definitions
2

 A data structure that consists of a set of
nodes (vertices) and a set of edges that
relate the nodes to each other
 The set of edges describes relationships
among the vertices. Some Examples,
 Car navigation system
 Efficient database
 Build a bot to retrieve info off WWW
 Representing computational models

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Applications
3

CS16
 electronic circuits

 networks (roads, flights,
communications)
JFK

LAX STL
HNL
DFW
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FTL

Classic Graph Theory Problems
4

Graph theory started from a mathematical curiosity.

"The Seven Bridges of Königsberg is a problem inspired
by an actual place and situation. The city of Kaliningrad,
Russia (at the time, Königsberg, Germany) is set on the
Pregolya River, and included two large islands which
were connected to each other and the mainland by seven
bridges. The question is whether it is possible to walk
with a route that crosses each bridge exactly once, and
return to the starting point. In 1736, Leonhard Euler
proved that it was not possible."
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Seven Bridges of Königsberg
5

"In proving the result, Euler formulated the problem in terms of
graph theory, by abstracting the case of Königsberg -- first, by
eliminating all features except the landmasses and the bridges
connecting them; second, by replacing each landmass with a dot,
called a vertex or node, and each bridge with a line, called an edge
or link. The resulting mathematical structure is called a graph."

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6

"The shape of a graph may be distorted in any way without
changing the graph itself, so long as the links between nodes are
unchanged. It does not matter whether the links are straight or
curved, or whether one node is to the left of another.

Euler realized that the problem could be solved in terms of the
degrees of the nodes. The degree of a node is the number of edges
touching it; in the Königsberg bridge graph, three nodes have
degree 3 and one has degree 5. Euler proved that a circuit of the
desired form is possible if and only if there are no nodes of odd
degree. Such a walk is called an Eulerian circuit or an Euler
tour. Since the graph corresponding to Königsberg has four
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nodes of odd degree, it cannot have an Eulerian circuit."

Euler‟s Theory
7

Euler path:
 A graph is said to be containing an Euler path if it
can be traced in 1 sweep without lifting the pencil
from the paper and without tracing the same edge
more than once. Vertices may be passed through
more than once. The starting and ending points need
not be the same.
Euler circuit:
 An Euler circuit is similar to an Euler path, except
that the starting and ending points must be the same.

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Euler‟s Theory
8

Graph Number of odd Number of even What does the path
vertices (vertices vertices (vertices contain?
connected to an connected to an
(Euler path = P;
odd number of even number of
edges) edges) Euler circuit = C;

Neither = N)

1 0 10 C

2 0 6 C

3 2 6 P

4 2 4 P

5 4 1 N

6 8 0 9/3/2012 N

Euler‟s Theory
9

 From the above table, we can observe that:
A graph with all vertices being even contains an
Euler circuit.
 A graph with 2 odd vertices and some even
vertices contains an Euler path.
 A graph with more than 2 odd vertices does not
contain any Euler path or circuit.

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10

"The problem can be modified to ask for a path
that traverses all bridges but does not have the
same starting and ending point. Such a walk is
called an Eulerian trail or Euler walk. Such a
path exists if and only if the graph has exactly two
nodes of odd degree, those nodes being the
starting and ending points. (So this too was
impossible for the seven bridges of Königsberg.)"
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Formal Definition:
11

 A graph, G=(V, E), consists of two sets:
 a finite non empty set of vertices(V), and

 a finite set (E) of unordered pairs of distinct vertices
called edges.
 V(G) and E(G) represent the sets of vertices and
edges of G, respectively.
 Vertex: In graph theory, a vertex (plural vertices) or
node or points is the fundamental unit out of which
graphs are formed.
 Edge or Arcs or Links: Gives the relationship between
the Two vertices.

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Examples for Graph

0 0 0

1 2 1 2
1
3
3 4 5 6
G1 2
G2
G3
V(G1)={0,1,2,3} E(G1)={(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)}
V(G2)={0,1,2,3,4,5,6} E(G2)={(0,1),(0,2),(1,3),(1,4),(2,5),(2,6)}
V(G3)={0,1,2} E(G3)={<0,1>,<1,0>,<1,2>}
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Graph Terminology
13

 Two vertices joined by an edge are called the
end vertices or endpoints of the edge.

 If an edge is directed its first endpoint is called
the origin and the other is called the
destination.

 Two vertices are said to be adjacent if they are
endpoints of the same edge.
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Graph Terminology
14

a B b h

d j
A D F
i
e
c

g
C
f

E

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Graph Terminology
15

a B b h

d j
A D F
i
e
c

g
C
Vertices A and B f
are endpoints of edge a
E

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Graph Terminology
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a B b h

d j
A D F
i
e
c

g
C
Vertex A is the f
origin of edge a
E

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Graph Terminology
17

a B b h

d j
A D F
i
e
c

g
C
Vertex B is the f
destination of edge a
E

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Graph Terminology
18

a B b h

d j
A D F
i
e
c

g
C
Vertices A and B are f
adjacent as they are
endpoints of edge a E

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Graph Terminology
19

 An edge is said to be incident on a vertex if the
vertex is one of the edges endpoints.

 The outgoing edges of a vertex are the
directed edges whose origin is that vertex.

 The incoming edges of a vertex are the
directed edges whose destination is that
vertex.
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Graph Terminology
20

a V b h

U d j
X i
Z
e
c

g Edge 'a' is incident on vertex V
W Edge 'h' is incident on vertex Z
Edge 'g' is incident on vertex Y
f

Y

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Graph Terminology
21

a V b h

U d j
X i
Z
e
c

g The outgoing edges of vertex W
W are the edges with vertex W as
origin {d, e, f}
f

Y

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Graph Terminology
22

a V b h

U d j
X i
Z
e
c

g The incoming edges of vertex X
W are the edges with vertex X as
destination {b, e, g, i}
f

Y

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23

Types of Graph

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Null graph, Trivial Graph
24

 A graph G=(V,E) where E=0 is said to be Null
or Empty graph v1

v2 v3

A graph with One vertex
and no edge is called as a trivial graph.

v1

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MultiGraph(Without Self Edge)
 The term multigraph refers to a graph in
which multiple edges between vertices are
permitted.
 A multigraph G = (V, E) is a graph which has
the set of vetrices and multiple edges between
vertices.
2

1 3

MultiGraph(With Self Edge)
 A multidigraph is a directed graph which is
permitted to have multiple edges, i.e., edges
with the own,source and target vertices.

2

1 3

Directed Graph
27

A directed graph is one in which every edge
(u, v) has a direction, so that (u, v) is different
from (v, u)
There are two possible situations that can arise
in a directed graph between vertices u and v.
 i) only one of (u, v) and (v, u) is present.

 ii) both (u, v) and (v, u) are present.

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Directed Graph
28

a V b h

U d j
X i
Z
e
c Here (u,v) is
possible g
where as
(v,u) is not W possible.
f

Y
In a directed edge, u is said to be adjacent to v and v is said
to be adjacent from u.
The edge <u,v> is incident to both u and v.
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Directed Graph
29

 Directed Graphs are also called as Digraph.
 Directed graph or the digraph make reference
to edges which are directed (i.e) edges which
are Ordered pairs of vertices.
 The edge(uv) is referred to as <u,v> which is
distinct from <v,u> where u,v are distinct
vertices.

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Undirected Graph
30

 In an undirected graph, there is no distinction
between (u, v) and (v, u).
• An edge (u, v) is said to be directed from u to v if
the pair (u, v) is ordered with u preceding v.
E.g. A Flight Route
• An edge (u, v) is said to be undirected if the pair
(u, v) is not ordered
E.g. Road Map
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Undirected Graph
31

a B b h

d j
A D F
i
e
c

g
C Here (u,v) and (v,u)
f both are possible.
E

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Undirected Graph
32

 A graph whose definition makes reference to
Unordered pairs of vertices as Edges is known
as undirected graph.
 Thus an undirected edge (u,v) is equivalent to
(v,u) where u and v are distinct vertices.
 In the case of undirected edge(u,v) in a graph,
the vertices u,v are said to be adjacent or the
edge(u,v) is said to be incident on vertices u,v.

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Complete Graph
33

 In a complete graph: Every node should be
connected to all other nodes.
 The above means “ Every node is adjacent to
all other nodes in that graph”.
 The degree of all the vertices must be same.

K1 K2 K3 K4 K5

 Kn = Denotes a complete with n number of
vertices. 9/3/2012

Complete Undirected Graph
34

An undirected graph with „n‟ number of vertices is
said to be complete ,iff each vertex

Number of vertices=3
v1 Degree of Each vertices
v2 =(n-1)
=(3-1)
=2
v3

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Complete Undirected Graph
35

 An n vertex undirected graph with exactly (n.(n-
1))/2 edges is said to be complete.
v1 Here we have 4
number of vertices and
hence
v2 v4 (4.(4-1))/2= (4.3)/2
=6

v3

Hence the graph has 6 number of edges and it is a
Complete Undirected graph.

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Complete Directed Graph
36

 An directed graph with „n‟ number of vertices is
said to be complete ,if each vertices has (n-1)
number of in-coming and out-going edges.
• In case of a digraph with n vertices, maximum
number of edges is given by n.(n-1).Such a graph
with exactly n.(n-1) edges is said to be Complete
digraph.

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Complete Directed Graph
37
Example:
Hence the graph has 6
number of edges and it is
v1 a Complete directed
graph.

v2

v3
 Here we have 3 number of vertices and hence
 n.(n-1)= 3.(3-1)
 =6

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Sub Graph
 A graph whose vertices and edges are subsets
of another graph.
 A subgraph G‟=(V‟,E‟) of a graph G = (V,E)
such that V‟ ⊆ V and E‟ ⊆ E , Then G is a
supergraph for G‟.

(G) (G1)

..Sub Graph

A B A
A B

C D C D
D

E
E

(G) (G1) (G2)

Spanning Subgraph
 A spanning subgraph is a subgraph that
contains all the vertices of the original graph.

Induced-Subgraph
 Vertex-Induced Subgraph:
 A vertex-induced subgraph is one that consists of
some of the vertices of the original graph and all of the
edges that connect them in the original.

A B B
C
C C

D E D E
D E

Induced-Subgraph

 Edge-Induced Subgraph:
 An edge-induced subgraph consists of some of the
edges of the original graph and the vertices that are at
their endpoints.
A A

B C B C
B

D E D E D

F F

Conclusions
43

Graph theory enables us to study and model networks and
solve some difficult problems inherently capable of being
modelled using networks.

Various terms e.g. vertex and edge, are associated with graph
theory which gives these terms special meanings. These
meanings need to be understood and remembered in order to
apply graph theoretic approaches to solving problems.

When solving a problem by developing a graph-based
program, careful attention must be given at the design stage to
the structuring of data to help make solving the problem
tractable, to enable linkages to be traced efficiently and to
avoid duplication of data.

Graph theory 1

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Graph theory 1