Graphs
Basic
Concepts
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Graphs – Basic Concepts
 Basic
definitions: vertices and edges
 More definitions: paths, simple paths,
cycles, loops
 Connected and disconnected graphs
 Complete graphs
 Weighted graphs
 Graph representation


Adjacency matrix
Adjacency lists
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Basic Graph Definitions
A graph is a non-linear data structure that consists of a
collection of nodes(or vertices) and a collection of edges(or
arcs), with each edge joining one node to another.
Formally, a graph G = (V,E) consists of two sets:
a) V called set of all vertices or nodes.
b) E called set of all edges or arcs.
Vertices are represented by circles or points and edges as
line segments or arcs connecting the vertices.
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Vertices and Edges
Definition: A graph is a collection (nonempty set)
of vertices and edges
Vertices: can have names
Edges:
connect two vertices,
can be labeled,
can be directed
Adjacent vertices: there is an edge between
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them
Graph1
Example
Vertices: A,B,C,D
Edges:
A
AB, AC, BC, CD
C
B
A
C
D
Two ways to draw the same graph
B
D
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Graph2
Directed and undirected
graphs
Graph3
A
B
A
B
C
C
D
D
These are two different graphs
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More definitions : Path
A list of vertices in which successive
vertices are connected by edges
ABC
A
B
BACD
ABCABCABCD
BABAC
C
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D
More definitions :
Simple Path
No vertex is repeated.
ABCD
D CA
A
B
C
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DCB
AB
ABC
D
More definitions : Cycle
A loop is a special case of cycle in which
an edge begin and ends with the same
vertex.
ABCA
A
B
BACB
CBAC
C
D
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A graph without cycles is called acyclic graph.
Connected and Disconnected
graphs
Connected graph: There is a path between each
two vertices
Disconnected graph : There are at least two
vertices not connected by a path.
Examples of disconnected graphs:
A
B
C
D
A
C
B
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D
Degree of vertex
In an undirected graph, the degree of a vertex is the
number of edges originating from it.
In directed graph degree of vertex comes in two
variations:
a) Indegree
b) Outdegree
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Complete graphs
Graphs with all edges present – each vertex is
connected to all other vertices
A
B
C
D
E
A complete graph
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Weighted graphs
Weighted graphs – weights are assigned to each
edge (e.g. road map)
1
B
2
C
2
A
4
3
D
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Graph Representation
 Adjacency matrix
 Adjacency lists
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Adjacency matrix – undirected
Vertices:
A,B,C,D
graphs
Edges:
AC, AB, AD, BD
The matrix is symmetrical
A
B
C
D
A
0
1
1
1
B
1
0
0
1
C
1
0
0
1
D
1
1
0
0
A
B
C
D
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Adjacency matrix – directed
Vertices:
A,B,C,D
graphs
Edges: AC, AB, BD, DA
A
B
C
D
A
0
1
1
0
B
0
0
0
1
C
0
0
0
0
D
1
0
0
0
A
B
C
D
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Adjacency lists – undirected
Vertices:
A,B,C,D
graphs
Edges:
AC, AB, AD, BD
Heads
lists
A
BCD
B
AD
C
A
D
AB
A
B
C
D
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Vertices:
Edges:
Adjacency lists – directed
graphs
A,B,C,D
AC, AB, BD, DA
A
Heads
lists
A
BC
B
D
C
=
D
A
B
C
D
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