graph theory

GRAPH THEORY
Prof S Sameen Fatima
Dept of Computer Science and Engineering
Osmania University College of Engineering
Hyderabad 500007
sameenf@gmail.com
Graph Theory S Sameen Fatima 1

OVERVIEW
• BASICS
• REPRESENTATION OF GRAPHS
• MINIMUM SPANNING TREE
• SEARCH ALGORITHMS
• EXAMPLES

BASICS
1. What Is a Graph?
2. Kinds of Graphs
3. Vertex Degree
4. Paths and Cycles

The KÖnigsberg Bridge Problem
• Königsber is a city on the Pregel river in
Prussia
• The city occupied two islands plus areas on
both banks
• Problem:
Whether they could leave home, cross every
bridge exactly once, and return home.
X
Y
Z
W

A Model
• A vertex : an island
• An edge : a path(bridge) between two
islands
e1
e2
e3
e4
e6
e5
e7
Z
Y
X
W
X
Y
Z
W

General Model
• A vertex : an object
• An edge : a relation between two objects
common
member
Committee 1 Committee 2

What Is a Graph?
• A graph G is an ordered pair (V, E)
consisting of:
– A vertex set V = {W, X, Y, Z}
– An edge set E = {e1, e2, e3, e4, e5, e6, e7}
e1
e2
e3
e4
e6
e5
e7
Z
Y
X
W

What Is a Graph?
• A graph, G is an ordered triple (V, E, f)
consisting of
– V is a set of nodes, points, or vertices.
– E is a set, whose elements are known as
edges or lines.
– f is a function that maps each element of E to
an unordered pair of vertices in V.

Loop, Multiple edges
• Loop : An edge whose endpoints are
equal
• Multiple edges : Edges have the same
pair of endpoints
loop
Multiple
edges

Simple Graph
Simple graph : A graph has no loops or multiple edges
loop
Multiple
edges
It is not simple. It is a simple graph.

Adjacent, neighbors
• Two vertices are adjacent and are
neighbors if they are the endpoints of an
edge
• Example:
– A and B are adjacent
– A and D are not adjacent
A B
C D

Finite Graph, Null Graph
• Finite graph : an graph whose vertex set
and edge set are finite
• Null graph : the graph whose vertex set
and edges are empty

Connected and Disconnected
• Connected : There exists at least one
path between two vertices
• Disconnected : Otherwise
• Example:
– H1 and H2 are connected
– H3 is disconnected
c
d
a b
de
a b
c
d
eH1
H3H2

Complete Graph
• Complete Graph: A simple graph in which every
pair of vertices are adjacent
• If no of vertices = n, then there are n(n-1) edges

Sparse/Dense Graph
• A graph is sparse if | E | | V |
• A graph is dense if | E | | V |2.

Directed Graph (digraph)
In a digraph edges have directions

Weighted Graph
Weighted graph is a graph for which each edge
has an associated weight, usually given by a
weight function w: E R.
1 2 3
4 5 6
.5
1.2
.2
.5
1.5
.3
1
4 5 6
2 3
2
1
35

Planar Graph
• Can be drawn on a plane such that no two edges intersect

Complement
Complement of G: The complement G’ of
a simple graph G :
– A simple graph
– V(G’) = V(G)
– E(G’) = { uv | uv E(G) }
G
u
v
w
x
y
G’
u
v
wx
y

Subgraphs
• A subgraph of a graph G is a graph H
such that:
– V(H) V(G) and E(H) E(G) and
– The assignment of endpoints to edges in H is
the same as in G.

Subgraphs
• Example: H1, H2, and H3 are subgraphs
of G
c
d
a b
de
a b
c
de
H1
G
H3
H2
a b
c
de

Bipartite Graphs
• A graph G is bipartite if V(G) is the union of
two disjoint independent sets called partite
sets of G
• Also: The vertices can be partitioned into
two sets such that each set is independent
• Matching Problem
• Job Assignment Problem
Workers
Jobs
Boys
Girls

Chromatic Number
• The chromatic number of a graph G,
written x(G), is the minimum number of
colors needed to label the vertices so that
adjacent vertices receive different colors
Red
Green
Blue
Blue
x(G) = 3

Maps and coloring
• A map is a partition of the plane into
connected regions
• Can we color the regions of every map
using at most four colors so that
neighboring regions have different
colors?
• Map Coloring graph coloring
– A region A vertex
– Adjacency An edge

Scheduling and Graph Coloring
• Two committees can not hold meetings
at the same time if two committees have
common member
common
member

• Model:
– One committee being represented by a
vertex
– An edge between two vertices if two
corresponding committees have common
member
– Two adjacent vertices can not receive the
same color
common
member

• Scheduling problem is equivalent to
graph coloring problem
Common
MemberCommittee 1
Committee 2
Committee 3
Common
Member
Different Color
No Common Member
Same Color OK
Same time slot OK

Degree
Degree: Number of edges incident on a node
A
D E F
B C
The degree of B is 2.

Degree (Directed Graphs)
• In degree: Number of edges entering a node
• Out degree: Number of edges leaving a node
• Degree = Indegree + Outdegree
1 2
4 5
The in degree of 2 is 2 and
the out degree of 2 is 3.

Degree: Simple Facts
• If G is a digraph with m edges, then
indeg(v) = outdeg(v) = m = |E |
• If G is a graph with m edges, then
deg(v) = 2m = 2 |E |
– Number of Odd degree Nodes is even

Path
• A path is a sequence of vertices such that there is
an edge from each vertex to its successor.
• A path is simple if each vertex is distinct.
• A circuit is a path in which the terminal vertex
coincides with the initial vertex
1 2 3
4 5 6
Simple path: [ 1, 2, 4, 5 ]
Path: [ 1, 2, 4, 5, 4]
Circuit: [ 1, 2, 4, 5, 4, 1]

Cycle
• A path from a vertex to itself is called a cycle.
• A graph is called cyclic if it contains a cycle;
– otherwise it is called acyclic
1 2 3
4 5 6
Cycle

Other Paths
• Geodesic path: shortest path
– Geodesic paths are not necessarily unique: It is quite possible to
have more than one path of equal length between a given pair of
vertices
– Diameter of a graph: the length of the longest geodesic path between
any pair of vertices in the network for which a path actually exists
• Eulerian path: a path that traverses each edge in a network exactly once
The Königsberg bridge problem
• Hamilton path: a path that visits each vertex in a network exactly once
34

Euclerian Path
• An undirected graph possesses an Euclerian Path
if and only if it is connected and has either zero or
two vertices of odd degree
OR
• An undirected graph possesses an Euclerian Path
if and only if it is connected and its vertices are all
of even degree
There is no Euclerian Path for the Konigsberg Bridge
Problem

GRAPH REPRESENTATION
• Adjacency Matrix
• Incidence Matrix
• Adjacency List

Adjacency, Incidence, and Degree
• Assume ei is an edge whose endpoints are (vj,vk)
• The vertices vj and vk are said to be adjacent
• The edge ei is said to be incident upon vj
• Degree of a vertex vk is the number of edges
incident upon vk . It is denoted as d(vk)
ei
vj vk

Adjacency Matrix
• Let G = (V, E), |V| = n and |E|=m
• The adjacency matrix of G written A(G), is the
|V| x |V| matrix in which entry ai,j is the
number of edges in G with endpoints {vi, vj}.
a
b
c
d
e
w
x
y z
w x y z
0 1 1 0
1 0 2 0
1 2 0 1
0 0 1 0
w
x
y
z

Adjacency Matrix
• Let G = (V, E), |V| = n and |E|=m
• The adjacency matrix of G written A(G), is the |V| x |V|
matrix in which entry ai,j is 1 if an edge exists otherwise it
is 0
1
5
2
4
3
1 2 3 4 5
1
2
3
4
5
0 1 0 0 1
1 0 1 1 1
0 1 0 1 0
0 1 1 0 1
1 1 0 1 0

Adjacency Matrix (Weighted Graph)
• Let G = (V, E), |V| = n and |E|=m
• The adjacency matrix of G written A(G), is the |V| x |V|
matrix in which entry ai,j is weight of the edge if it exists
otherwise it is 0
1
5
2
4
3
1 2 3 4 5
1
2
3
4
5
0 5 0 0 1
5 0 4 6 3
0 4 0 2 0
0 6 2 0 7
1 3 0 7 0
5
3
7
61
2
4

Incidence Matrix
• Let G = (V, E), |V| = n and |E|=m
• The incidence matrix M(G) is the |V| x |E|
matrix in which entry mi,j is 1 if vi is an endpoint
of ei and otherwise is 0.
a
b
c
d
e
w
x
y
z
a b c d e
1 1 0 0 0
1 0 1 1 0
0 1 1 1 1
0 0 0 0 1
w
x
y
z

Adjacency List Representation
• Adjacency-list representation
– an array of |V | elements, one for each vertex in V
– For each u V , ADJ [ u ] points to all its adjacent
vertices.

Adjacency List Representation
for a Digraph
1
5
1
22
5
4 4
3 3
2 5
5 3 4
4
5
5

Minimum Spanning Tree

Minimum Spanning Tree
• What is MST?
• Kruskal's Algorithm
• Prim's Algorithm

A spanning tree of a graph is just a subgraph that
contains all the vertices and is a tree.
A graph may have many spanning trees.
o
r
o
r
o
r
Some Spanning Trees from Graph AGraph A
Spanning Trees

All 16 of its Spanning TreesComplete Graph

Minimum Spanning Trees
The Minimum Spanning Tree for a given graph is the Spanning Tree of minimum cost for that
graph.
5
7
2
1
3
4
2
1
3
Complete Graph Minimum Spanning Tree

Kruskal's Algorithm
This algorithm creates a forest of trees. Initially the forest consists of n single
node trees (and no edges). At each step, we add one edge (the cheapest one)
so that it joins two trees together. If it were to form a cycle, it would simply
link two nodes that were already part of a single connected tree, so that this
edge would not be needed.

The steps are:
1. The forest is constructed - with each node in a separate tree.
2. The edges are placed in a priority queue.
3. Until we've added n-1 edges,
1. Extract the cheapest edge from the queue,
2. If it forms a cycle, reject it,
3. Else add it to the forest. Adding it to the forest will join two trees together.
Every step will have joined two trees in the forest together, so that at the end,
there will only be one tree in T.

4
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
Complete Graph

1
4
2
5
2
5
4
3
4
4
4
10
1
6
3
3
2
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
A AB D
B B
B
C D
J C
C
E
F
D
D H
J E G
F FG I
G GI J
H J JI

2
5
2
5
4
3
4
4
10
1
6
3
3
2
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
B
B
D
J
C
C
E
F
D
D H
J
E G
F
F
G
I
G
G
I
J
H J
JI
1A D
4B C
4A B
Sort Edges
(in reality they are placed in a priority
queue - not sorted - but sorting them
makes the algorithm easier to visualize)

2
5
2
5
4
3
4
4
10
1
6
3
3
2
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
B
B
D
J
C
C
E
F
D
D H
J
E G
F
F
G
I
G
G
I
J
H J
JI
1A D
4B C
4A B
Add Edge

2
5
2
5
4
3
4
4
10
1
6
3
3
2
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
B
B
D
J
C
C
E
F
D
D H
J
E G
F
F
G
I
G
G
I
J
H J
JI
1A D
4B C
4A B
Cycle
Don’t Add Edge

2
5
2
5
4
3
4
4
10
1
6
3
3
2
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
B
B
D
J
C
C
E
F
D
D H
J
E G
F
F
G
I
G
G
I
J
H J
JI
1A D
4B C
4A B
Add Edge

2
5
2
5
4
3
4
4
10
1
6
3
3
2
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
B
B
D
J
C
C
E
F
D
D H
J
E G
F
F
G
I
G
G
I
J
H J
JI
1A D
4B C
4A B
Cycle
Don’t Add Edge

2
5
2
5
4
3
4
4
10
1
6
3
3
2
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
B
B
D
J
C
C
E
F
D
D H
J
E G
F
F
G
I
G
G
I
J
H J
JI
1A D
4B C
4A B
Add Edge

4
1
2
2 1
3
32
4
A
B C
D
E F
G
H
I
J
4
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
Minimum Spanning Tree Complete Graph

Prim's Algorithm
This algorithm starts with one node. It then, one by one, adds a node that
is unconnected to the new graph, each time selecting the node whose
connecting edge has the smallest weight out of the available nodes’
connecting edges.

The steps are:
1. The new graph is constructed - with one node from the old graph.
2. While new graph has fewer than n nodes,
1. Find the node from the old graph with the smallest connecting
edge to the new graph,
2. Add it to the new graph
Every step will have joined one node, so that at the end we will have one
graph with all the nodes and it will be a minimum spanning tree of the
original graph.

4
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
Complete Graph

4
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J
Old Graph New Graph
4
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J

4
1
2
2 1
3
32
4
A
B C
D
E F
G
H
I
J
Complete Graph Minimum Spanning Tree
4
1
2 3
2 1
3
5
3
4
2
5 6
4
4
10
A
B C
D
E F
G
H
I
J

Examples
• Cost of wiring electronic components
• Shortest route between two cities.
• Shortest distance between all pairs of cities in
a road atlas.
• Matching / Resource Allocation
• Task scheduling

Examples
• Flow of material
– liquid flowing through pipes
– current through electrical networks
– information through communication networks
– parts through an assembly line
• In Operating systems to model resource handling
(deadlock problems)
• In compilers for parsing and optimizing the code.

Graph Algorithms

Search Algorithms
• Breadth First Search
• Depth Dirst Search

Breadth-First Search(BFS)
1. open  (initial state).
2. If open is empty , report failure , stop.
3. s  pop ( open )
4. If s is a solution , report s, stop.
5. succs  successors(s).
6. Add succs to tail of open.
7. go to 2.

Space for BFS
Space is calculated in terms of open list.
In the worst case:
The solution may be the rightmost node at the last level
At the last level (level d) the no. of nodes = bd
Therefore,
Total no: of nodes in the open list = bd
Space O (bd)

Time for BFS
In the worst case, the solution will be the right most node at depth
d, that is all the nodes would be expanded upto depth d.
No. of nodes processed at 1st level = 1
No. of nodes processed at the 2nd level = b
No: of nodes processed at the 3rd level = b2
…………..
No. of nodes processed at the dth level = bd
Therefore,
Total no. of nodes processed = 1 + b + b2 +……..bd.
= b ( bd – 1 )
(b – 1 )
O ( bd )
(ignoring 1 in comparison to b)

Depth-First Search(DFS)
1. open  ( initial state )
2. If open is empty , report failure ,stop.
3. s  pop ( open )
4. If s is a solution, report s, stop.
5. succs  successors (s).
6. add succs to head of open
7. go to 2.

Space for DFS
Space is calculated in terms of open list.
In the worst case:
At the last level (level d) the no. of nodes = b
At each of the preceding (d-1) levels i.e., 1, 2, 3, …., (d-1), the no. of nodes = b-1
Therefore,
Total no: of nodes at the preceding (d-1) levels = (d-1)(b-1)
Space b+ (d-1) ( b-1)
b + db – d – b + 1
d ( b – 1) + 1
bd
O ( d )
( In terms of open list )

Time for DFS
In the worst case, the solution will be the right most node at depth
d, that is all the nodes would be expanded upto depth d.
No. of nodes processed at 1st level = 1
No. of nodes processed at the 2nd level = b
No: of nodes processed at the 3rd level = b2
…………..
No. of nodes processed at the dth level = bd
Therefore,
Total no. of nodes processed = 1 + b + b2 +……..bd.
= b ( bd – 1 )
(b – 1 )
O ( bd )
(ignoring 1 in comparison to b)

Depth-First Iterative Deepening
(DFID)
The depth-first iterative deepening algorithm combines the
advantage of low space requirement of depth first search (DFS) and
advantage of finding an optimal solution of the breadth first search
(BFS)
time requirement, which is the same for both BFS and DFS
1. d  1
2. result  depth first (initial state, d)
3. (Comment: try to find a solution of length d using depth first
search)
4. If result ≠ NIL, report it, stop
5. d  d+ 1
6. go to 2

Time requirement for DFID
On a search to depth d =1, b nodes are visited. For d = 2 , b1 + b2 nodes are visited and so on.
At d = k, b1 + b2 + …….. + bk nodes are visited
Let’s define the cost of DFID as a recurrence relation
DFID(1) = b1
DFID(k) = + DFID(k-1)
This expands to
bk + bk-1 + bk-2 + ………………….. + b1
bk-1 + bk-2 + ………………….. + b1
bk-2 + ………………….. + b1
……
…….
……..
___________________________________________________________________
bk +2bk-1 +3bk-2 + ………………….. +kb1
= bk
For large k the above expression asymptotes to bk (as the expression in the parenthesis
asymptotes to 1). Hence time required for DFID is O(bk)

Examples
• Shortest route between two cities.
• Shortest distance between all pairs of cities in
a road atlas.
• Matching / Resource Allocation
• Task scheduling
• Cost of wiring electronic components
• Visibility / Coverage

Examples
• Flow of material
– liquid flowing through pipes
– current through electrical networks
– information through communication networks
– parts through an assembly line
• In Operating systems to model resource handling
(deadlock problems)
• In compilers for parsing and optimizing the code.

Thank you

graph theory

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