Graphs: Definition, Applications, and Terminology, Slides of Data Structures and Algorithms

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Data Structures & Algorithm

Analysis

What is a Graph?

• A graph G = (V,E) is composed of: V: set of vertices E: set of edges connecting the vertices in V
• An edge e = (u,v) is a pair of vertices
• Example:

a b

c

d e

V= {a,b,c,d,e}

E= {(a,b),(a,c),(a,d), (b,e),(c,d),(c,e), (d,e)}

Terminology:

Adjacent and Incident

• If (v 0 , v 1 ) is an edge in an undirected graph,
  • v 0 and v 1 are adjacent
  • The edge (v 0 , v 1 ) is incident on vertices v 0 and v 1
• If <v 0 , v 1 > is an edge in a directed graph
  • v 0 is adjacent to v 1 , and v 1 is adjacent from v 0
  • The edge <v 0 , v 1 > is incident on v 0 and v 1

The degree of a vertex is the number of edges

incident to that vertex

For directed graph,

the in-degree of a vertex v is the number of edges that have v as the head the out-degree of a vertex v is the number of edges that have v as the tail if di is the degree of a vertex i in a graph G with n vertices and e edges, the number of edges is

e d i

n

−

0

1

Terminology:

Degree of a Vertex

Why? Since adjacent vertices each count the adjoining edge, it will be counted twice

Terminology:

Path

• path: sequence of vertices v 1 ,v 2 ,.. .v (^) k such that consecutive vertices vi and v (^) i+1 are adjacent.

7

3

(^3 )

3

2

a (^) b

c

d e

a (^) b

c

d e a b e d c b e d c

More Terminology

• simple path: no repeated vertices
• cycle: simple path, except that the last vertex is the same as the first vertex

a (^) b

c

d e

b e c

a c d a

a (^) b

c

d e

(i) (ii) (iii) (^) 3 (iv) (a) Some of the subgraph of G (^1)

0 0

(i) (ii) (iii) (iv) (b) Some of the subgraph of G (^3)

G 1

G 3

Subgraphs Examples

More…

• tree - connected graph without cycles
• forest - collection of trees

tree

forest

tree

tree

tree

More Connectivity

n = #vertices

m = #edges

• For a tree m = n - 1

n = 5

m = 4

n = 5

m = 3

If m < n - 1, G is not connected

Oriented (Directed) Graph

• A graph where edges are directed

ADT for Graph

objects: a nonempty set of vertices and a set of undirected edges, where each edge is a pair of vertices functions: for all graph ∈ Graph , v , v 1 and v 2 ∈ Vertices Graph Create()::=return an empty graph Graph InsertVertex( graph , v )::= return a graph with v inserted. v has no incident edge. Graph InsertEdge( graph , v 1 , v 2 )::= return a graph with new edge between v 1 and v 2 Graph DeleteVertex( graph , v )::= return a graph in which v and all edges incident to it are removed Graph DeleteEdge( graph , v 1 , v 2 )::=return a graph in which the edge ( v 1 , v 2 ) is removed Boolean IsEmpty( graph )::= if ( graph == empty graph ) return TRUE else return FALSE List Adjacent( graph , v )::= return a list of all vertices that are adjacent to v

Graph Representations

Adjacency Matrix

Adjacency Lists

Examples for Adjacency Matrix

0 1 1 1

1 0 1 1

1 1 0 1

1 1 1 0





   





   

0 1 0

1 0 0

0 1 0





  





   0 1 1 0 0 0 0 0

1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 



          





          

G 1

G 2

G 4

symmetric

undirected: n 2 / directed: n 2 Docsity.com

Merits of Adjacency Matrix

From the adjacency matrix, to determine the

connection of vertices is easy

The degree of a vertex is

For a digraph (= directed graph), the row sum is

the out_degree, while the column sum is the

in_degree

adj mat i j

j

n

_ [ ][ ]

=

−

0

1

ind vi A j i j

n ( ) = [ , ] =

− ∑ 0

1 outd vi A i j j

n ( ) = [ , ] =

− ∑ 0

1