Graph Concepts
Samia Qader
Hw # 5
all images obtained from LEDA software in the following directory:
252a/handout/demo/graphwin/graphwin
Nodes, Edges
• Graph G = (V,E) where
V = (1,2, ..,n)
n= no.of nodes or vertices
E= the edge set of G, its
elements are called edges
• Examples of nodes:
0,1,2,3
• Examples of Edges:
between 0,1 and 0,2
Multiple Edges
• When there is
more than one edge
between a set of
vertices this is
known as a multiple
edge.
• Example: Multiple
edge between
nodes 2 and 5
Loops
• A loop is an edge
leaving from one
vertex(node),
returning to the
same vertex.
• Example: loop on
node 3
Undirected Graph
• An undirected
G= (V,E), the edge
set E consists of
unordered pair of
vertices.
• Self-loops are
forbidden in an
undirected graph
Directed Graph (digraph)
• A directed graph
or digraph G is a
pair (V,E) such that
the edge set E has
an ordered pair of
vertices.
Simple Graph
• A simple graph has
no loops and no
multiple edges
Examples of graphs
• Graph G = (V,E)
where
V = (1,2, ..,n)
n= no.of nodes or
vertices
E= the edge set of
G, its elements are
called edges
Examples of multigraphs
• A multigraph is like
an undirected
graph but it can
have both multiple
edges between
vertices and selfloops
A Complete Graph
• A complete graph is
an undirected
graph in which
every pair of
vertices is
adjacent (i.e. there
is an edge between
every pair of
vertices)
A bipartite graph
• In a bipartite
graph, the vertices
are divided into
two classes and all
the edges go
between these two
classes of vertices.
A path in an undirected graph
• A path in an undirected
graph from vertex a to
vertex z is a sequence
of vertices (v0,v1,..,vk)
s.t. a = v0 and z= vk.
• The length of the path
is the number of edges
in the path
• Example: path from 0
to 5 is of length 5.
Path in a directed graph
• A path in an undirected
graph from vertex a to
vertex z is a sequence
of vertices (v0,v1,..,vk)
s.t. a = v0 and z= vk.
• The length of the path
is the number of edges
in the path
• Example: path from 0 to
5 is of length 5.
Hamilton Path
• Hamilton path in a
graph is a path
that spans all
vertices
• Example 1: In a
directed graph
• Example 2: In an
undirected graph
Example 1
Example 2
A cycle in a directed graph
• In a directed graph, a
path (v0,v1,…,vk)
forms a cycle if v0=vk
and the path contains
at least one edge
• In a cycle, each
vertex has exactly 2
neighbors and the
graph is connected
A cycle in an undirected
graph
• In an undirected
graph, a path (v0,
v1, …,vk) forms a
cycle if v0 = vk and
v0,v1,..,vk are
distinct
• Example: nodes
0,1,2,3 form a cycle
Hamilton Cycle
Example 1
• A Hamilton Cycle is
a cycle that covers
all the vertices
• Example 1: A
Hamilton Path in a Example 2
digraph
• Example 2: A
Hamilton Path in an
undirected graph
Cyclic and Acyclic Digraphs
• A cyclic digraph
contains cyclesexample 1
• An acyclic digraph
has no cyclesexample 2
Example 1
Example 2
Tree
• A tree is a special
class of graphs. It
is a connected,
acyclic, undirected
graph.
Forest
• A group of trees
that is not
connected is called
a forest
• If an undirected
graph is acyclic but
possibly
disconnected, it is
a forest
Graph which is not connected
• An undirected graph
is not connected if
every pair of vertices
is not connected by a
path.
• Example1: This graph
is not connected
because vertex 2 is
not connected by a
path.
Example 1
Graph which is not connected
Example 2
• Example 2: This
graph is not
connected since
vertices 2 and 8
are not connected
by a path.