2. Applications of Graphs
Potentially anything (graphs can represent
relations, relations can describe the extension
of any predicate)
Applications in networking, scheduling, flow
optimization, circuit design, path planning.
Geneology analysis, computer game playing,
program compilation, object-oriented
design,….
3. Varying Applications (examples)
Computer networks
Distinguish between two chemical compounds
with the same molecular formula but different
structures
Solve shortest path problems between cities
Scheduling exams and assign channels to
television stations
5. Definitions - Graph
A generalization of the simple concept of a set
of dots, links, edges or arcs.
Representation: Graph G =(V, E) consists set of vertices
denoted by V, or by V(G) and set of edges E, or E(G)
6. Example
What is a graph G?
It is a pair G = (V, E),
where
V = V(G) = set of vertices
E = E(G) = set of edges
Example:
V = {s, u, v, w, x, y, z}
E = {(x,s), (x,v)1, (x,v)2, (x,u),
(v,w), (s,v), (s,u), (s,w), (s,y),
(w,y), (u,y), (u,z),(y,z)}
7. Edges
An edge may be labeled by a pair of
vertices, for instance e = (v,w).
e is said to be incident on v and w.
Isolated vertex = a vertex without
incident edges.
8. Definitions – Edge Type
Directed: Ordered pair of vertices. Represented as (u, v)
directed from vertex u to v.
Undirected: Unordered pair of vertices. Represented as {u, v}.
Disregards any sense of direction and treats both end vertices
interchangeably.
u v
u v
9. Definitions – Edge Type
Loop: A loop is an edge whose endpoints are equal
i.e., an edge joining a vertex to it self is called a loop.
Represented as {u, u} = {u}
Multiple Edges: Two or more edges joining the
same pair of vertices.
u
10. Definitions – Graph Type
Simple (Undirected) Graph: consists of V, a nonempty set
of vertices, and E, a set of unordered pairs of distinct elements
of V called edges (undirected)
Representation Example: G(V, E), V = {u, v, w}, E = {{u, v},
{v, w}, {u, w}}
u v
w
11. Definitions – Graph Type
Multigraph: G(V,E), consists of set of vertices V, set of
Edges E and a function f from E to {{u, v}| u, v V, u ≠ v}.
The edges e1 and e2 are called multiple or parallel edges if f
(e1) = f (e2).
Representation Example: V = {u, v, w}, E = {e1, e2, e3}
e2
u
v
w
e1
e3
• E.g., nodes are cities, edges are segments of major highways.
12. Definitions – Graph Type
Pseudograph: Like a multigraph, but edges connecting a
node to itself are allowed.
Representation Example: V = {u, v, w}, E = {e1, e2, e3, e4}
u
v
w
e1
e3
e2
e4
13. Definitions – Graph Type
Directed Graph: G(V, E), set of vertices V, and set of Edges E,
that are ordered pair of elements of V (directed edges)
Representation Example: G(V, E), V = {u, v, w}, E = {(u, v), (v,
w), (w, u)}
u
w
v
E.g. : V = people , E={( ={(x, y) | ) | x likes y}
14. Definitions – Graph Type
Directed Multigraph: Like directed graphs, but there may be
more than one arc from a node to another.
Representation Example: V = {u, v, w}, E = {e1, e2, e3, e4}
u
w
v
e1
e2
e3
e4
15. Definitions – Graph Type
Type Edges Multiple Edges
Allowed ?
Loops Allowed ?
Simple Graph undirected No No
Multigraph undirected Yes No
Pseudograph undirected Yes Yes
Directed Graph directed No Yes
Directed
Multigraph
directed Yes Yes
16. Terminology – Undirected graphs
u and v are adjacent if {u, v} is an edge, e is called incident
with u and v. u and v are called endpoints of {u, v}
Degree of Vertex (deg (v)): the number of edges incident
on a vertex. A loop contributes twice to the degree (why?).
Pendant Vertex: deg (v) =1
Isolated Vertex: deg (v) = 0
Representation Example: For V = {u, v, w} , E = { {u, w}, {u,
w}, (u, v) }, deg (u) = 2, deg (v) = 1, deg (w) = 1, deg (k) =
0, w and v are pendant , k is isolated
u
k
w
v
17. Terminology – Directed graphs
For the edge (u, v), u is adjacent to v OR v is adjacent from
u, u – Initial vertex, v – Terminal vertex
In-degree (deg- (u)): number of edges for which u is
terminal vertex
Out-degree (deg+ (u)): number of edges for which u is initial
vertex
Note: A loop contributes 1 to both in-degree and out-degree
(why?)
Representation Example: For V = {u, v, w} , E = { (u, w), ( v,
w), (u, v) }, deg- (u) = 0, deg+ (u) = 2, deg- (v) = 1,
deg+ (v) = 1, and deg- (w) = 2, deg+ (u) = 0
u
w
v
19. Theorems: Undirected Graphs
Theorem 2:
An undirected graph has even number of vertices with odd
degree
even
V
oof
2
2
1
V
v
1,
V
v
V
u
V
v
deg(v)
term
second
even
also
is
term
second
Hence
2e.
is
sum
since
even
is
inequality
last
the
of
side
hand
right
on the
terms
last two
the
of
sum
The
even.
is
inequality
last
the
of
side
hand
right
in the
first term
The
V
for v
even
is
(v)
deg
deg(v)
deg(u)
deg(v)
2e
vertices
degree
odd
to
refers
V2
and
vertices
degree
even
of
set
the
is
1
Pr
21. Simple graphs – special cases
Complete graph: Kn, is the simple graph that contains exactly
one edge between each pair of distinct vertices.
Representation Example: K1, K2, K3, K4
K2
K1
K4
K3
22. Simple graphs – special cases
Cycle: Cn, n ≥ 3 consists of n vertices v1, v2, v3 … vn and edges
{v1, v2}, {v2, v3}, {v3, v4} … {vn-1, vn}, {vn, v1}
Representation Example: C3, C4
C3 C4
How many edges are there in Cn ?
23. Simple graphs – special cases
Wheels: Wn, obtained by adding additional vertex to Cn and
connecting all vertices to this new vertex by new edges.
Representation Example: W3, W4
W3 W4
How many edges are there in Wn?
24. Simple graphs – special cases
N-cubes: Qn, vertices represented by 2^n bit strings of length
n. Two vertices are adjacent if and only if the bit strings that
they represent differ by exactly one bit positions
Representation Example: Q1, Q2
0 1
Q1
10
00
11
01
Q2
What is Q3?
25. Bipartite graphs
In a simple graph G, if V can be partitioned into two disjoint
sets V1 and V2 such that every edge in the graph connects a
vertex in V1 and a vertex V2 (so that no edge in G connects
either two vertices in V1 or two vertices in V2)
Application example: Representing Relations
Representation example: V1 = {v1, v2, v3} and V2 = {v4, v5, v6},
v1
v2
v3
v4
v5
v6
V1
V2
V(G) = V(V1) V(V2)
|V(V1)| = m, |V(V2)| = n
V(V1) V(V2) =
26. Complete Bipartite graphs
Km,n is the graph that has its vertex set portioned into two
subsets of m and n vertices, respectively There is an edge
between two vertices if and only if one vertex is in the first
subset and the other vertex is in the second subset.
Representation example: K2,3, K3,3
K3,3
|V(V1)| = m
|V(V2)| = n
K2,3
V1
V2
28. Subgraphs
A subgraph of a graph G = (V, E) is a graph H =(V’, E’)
where V’ is a subset of V and E’ is a subset of E
Application example: solving sub-problems within a graph
Representation example: V = {u, v, w}, E = ({u, v}, {v, w},
{w, u}}, H1 , H2
u
v
H2
u
v w
G
u
w
v
H1
29. Subgraphs
G = G1 U G2 wherein E = E1 U E2 and V = V1 U V2, G, G1
and G2 are simple graphs of G
Representation example:
V1 = {u, w}, E1 = {{u, w}}, V2 = {w, v}, E1 = {{w, v}},
V = {u, v ,w}, E = { { {u, w}, {w, v} } }
u
v
w w
v
w
u
G1 G2 G
30. Representation
Incidence (Matrix): Most useful when
information about edges is more desirable than
information about vertices.
Adjacency (Matrix/List): Most useful when
information about the vertices is more desirable
than information about the edges. These two
representations are also most popular since
information about the vertices is often more
desirable than edges in most applications
31. Representation- Incidence matrix
Incidence matrix
Label rows with vertices
Label columns with
edges
1 if an edge is incident
to a vertex, 0 otherwise
e f g h j
v 1 1 0 0 0
w 1 0 1 0 1
x 0 0 0 1 1
y 0 1 1 1 0
32. Representation- Adjacency Matrix
There is an N x N matrix, where |V| = N , the Adjacenct Matrix
(NxN) A = [aij]
For undirected graph
For directed graph
This makes it easier to find subgraphs, and to reverse graphs
if needed.
otherwise
0
G
of
edge
an
is
)
v
,
(v
if
1
a
j
i
ij
otherwise
0
G
of
edge
an
is
}
v
,
{v
if
1
a
j
i
ij
33. Representation- Adjacency Matrix
Adjacency is chosen on the ordering of
vertices. Hence, there as are as many as n!
such matrices.
The adjacency matrix of simple graphs are
symmetric (aij = aji) (why?)
When there are relatively few edges in the
graph the adjacency matrix is a sparse
matrix
Directed Multigraphs can be represented by
using aij = number of edges from vi to vj
36. Linked Representation-
Adjacency List
Each node (vertex) has a list of which nodes (vertex) it is adjacent
Example: undirectd graph G (V, E)
u
v w
node Adjacency List
u v , w
v w, u
w u , v
G= [u: v, w; v: w, u; w: u, v]
37. Graph - Isomorphism
G1 and G2 are isomorphic
if there exist one-to-one onto functions f:
V(G1) → V(G2) and g: E(G1) → E(G2) such
that
an edge e is adjacent to vertices v, w in G1 if
and only if g(e) is adjacent to f(v) and f(w) in
G2
39. Connectivity
Basic Idea: In a Graph Reachability among vertices
by traversing the edges
Application Example:
- In a city to city road-network, if one city can be
reached from another city.
- Problems if determining whether a message can be
sent between two computer using intermediate links
- Efficiently planning routes for data delivery in the
Internet
40. Connectivity – Path
A Path is a sequence of edges that begins at a
vertex of a graph and travels along edges of the
graph, always connecting pairs of adjacent
vertices.
Representation example: G = (V, E), Path P represented,
from u to v is
{{u, 1}, {1, 4}, {4, 5}, {5, v}}
1
u
3
4 5
2
v
41. Connectivity – Path
Definition for Directed Graphs
A Path of length n (> 0) from u to v in G is a sequence of n
edges e1, e2 , e3, …, en of G such that f (e1) = (xo, x1), f (e2) =
(x1, x2), …, f (en) = (xn-1, xn), where x0 = u and xn = v. A path is
said to pass through x0, x1, …, xn or traverse e1, e2 , e3, …, en
For Simple Graphs, sequence is x0, x1, …, xn
In directed multigraphs when it is not necessary to distinguish
between their edges, we can use sequence of vertices to
represent the path
Circuit/Cycle: u = v, length of path > 0
Simple Path: does not contain an edge more than once
42. Paths and cycles
A path of length n is
a sequence of n + 1
vertices and n
consecutive edges
A cycle is a path that
begins and ends at
the same vertex
43. Connectivity – Connectedness
Undirected Graph
An undirected graph is connected if there exists is a
simple path between every pair of vertices
Representation Example: G (V, E) is connected since
for V = {v1, v2, v3, v4, v5}, there exists a path
between {vi, vj}, 1 ≤ i, j≤ 5
v1
v2
v3
v5
v4
44. Connectivity – Connectedness
Undirected Graph
Articulation Point (Cut vertex): removal of a vertex
produces a subgraph with more connected components than
in the original graph. The removal of a cut vertex from a
connected graph produces a graph that is not connected
Cut Edge: An edge whose removal produces a subgraph
with more connected components than in the original graph.
Representation example: G (V, E), v3 is the articulation point
or edge {v2, v3}, the number of connected components is 2
(> 1)
v1
v2
v3
v4
v5
45. Connectivity – Connectedness
Directed Graph
A directed graph is strongly connected if there is a
path from a to b and from b to a whenever a and b are
vertices in the graph.
A directed graph is weakly connected if there is a
(undirected) path between every two vertices in the
underlying undirected path
A strongly connected Graph can be weakly connected but
the vice-versa is not true (why?)
a b
c d
a b
c d
46. Unilaterally connected:
A directed graph is unilaterally connected or unilateral
if for any pair of vertices a and b, there is a path from
a to b to b to a, i.e. one of them is reachable from the
other.
Strongly connected=> Unilaterally=>Weakly connected
Digraph G Digraph H
Verify that G is strongly connected. H is not strongly connected.
Is H unilateral ?
47. Connectivity – Connectedness
Directed Graph
Representation example: G1 (Strong component), G2 (Weak
Component), G3 is undirected graph representation of G2 or G1
G2
G1 G3
48. Connectivity – Connectedness
Directed Graph
Strongly connected Components: subgraphs of a
Graph G that are strongly connected
Representation example: G1 is the strongly
connected component in G
G1
G
49. Counting Paths
Theorem: Let G be a graph with adjacency matrix A
with respect to the ordering v1, v2, …, Vn (with directed
on undirected edges, with multiple edges and loops
allowed). The number of different paths of length r from
Vi to Vj, where r is a positive integer, equals the (i, j)th
entry of (adjacency matrix) Ar.
Proof: By Mathematical Induction.
50. Counting Paths
Theorem: Let G be a graph with …. of (adjacency matrix) Ar.
Proof: By Mathematical Induction.
Base Case: For the case N = 1, aij =1 implies that there
is a path of length 1. This is true since this corresponds
to an edge between two vertices.
We assume that theorem is true for N = r and prove the
same for N = r +1. Assume that the (i, j)th entry of Ar is
the number of different paths of length r from vi to vj. By
induction hypothesis, bik is the number of paths of
length r from vi to vk.
51. Counting Paths
Case r +1: In Ar+1 = Ar. A,
The (i, j)th entry in Ar+1 , bi1a1j + bi2 a2j + …+ bin anj
where bik is the (i, j)th entry of Ar.
By induction hypothesis, bik is the number of paths of
length r from vi to vk.
The (i, j)th entry in Ar+1 corresponds to the length between
i and j and the length is r+1. This path is made up of
length r from vi to vk and of length from vk to vj. By product
rule for counting, the number of such paths is bik* akj The
result is bi1a1j + bi2 a2j + …+ bin anj ,the desired result.
52. Counting Paths
a ------- b
| |
| |
c -------d
A = 0 1 1 0 A4 = 8 0 0 8
1 0 0 1 0 8 8 0
1 0 0 1 0 8 8 0
0 1 1 0 8 0 0 8
Number of paths of length 4 from a to d is (1,4) th entry of A4 = 8.
53. Path Matrix:
Let G be a simple directed graph with m vertices
v1, v2,…,vm. The path matrix or reachability matrix
of G is the square matrix P = [pij]m×m defined as
otherwise
pij
0
v
to
v
from
path
a
is
there
if
1 j
i
Theorem: Let A be the adjacency matrix of a graph G
with m vertices and let Bm= A + A2 + A3 + …+Am.
Then the path matrix P and Bm have the same
nonzero entries.
54. Remarks:
In a strongly connected directed graph, for any pair
of vertices u and v in G, there is a path from u to v
and from v to u => G is Strongly connected if and
only if the path matrix P of G has no zero entries.
Consider the following digraph:
56. Continued:
By the above
theorem, the path
matrix P will be
given by
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
P
The whole zero column represents that v2 is not
Reachable from any other vertices. Since the matrix
P has zero entries therefore the digraph is not
strongly connected.
58. The Seven Bridges of Königsberg, Germany
The residents of Königsberg wondered if it was
possible to take a walking tour of the town that
crossed each of the seven bridges over the Presel river
exactly once. Is it possible to start at some node and
take a walk that uses each edge exactly once, and
ends at the starting node?
59. The Seven Bridges of Königsberg
You can redraw the original picture as long as for every edge between
nodes i and j in the original you put an edge between nodes i and j in
the redrawn version (and you put no other edges in the redrawn
version).
Original:
2
3
4 1
Redrawn:
4
2 3
60. The Seven Bridges of Königsberg
Has no tour that uses each edge exactly once.
(Even if we allow the walk to start and finish in different
places.)
Can you see why?
Euler:
61. Euler - definitions
An Eulerian path (Eulerian trail, Euler walk) in a graph
is a path that uses each edge precisely once. If such a path
exists, the graph is called traversable.
An Eulerian cycle (Eulerian circuit, Euler tour) in a
graph is a cycle that uses each edge precisely once. If such
a cycle exists, the graph is called Eulerian (also
unicursal).
Representation example: G1 has Euler path a, c, d, e, b, d,
a, b
a b
c d e
62. The problem in our language:
Show that is not Eulerian.
In fact, it contains no Euler trail.
63. Euler - theorems
1. A connected graph G is Eulerian if and only
if G is connected and has no vertices of odd
degree
2. A connected graph G is has an Euler trail
from node a to some other node b if and
only if G is connected and a b are the only
two nodes of odd degree
64. Euler – theorems (=>)
Assume G has an Euler trail T from node a to node b (a and b
not necessarily distinct).
For every node besides a and b, T uses an edge to exit for each
edge it uses to enter. Thus, the degree of the node is even.
1. If a = b, then a also has even degree. Euler circuit
2. If a b, then a and b both have odd degree. Euler path
65. Theorems
Theorem: If G is a graph with m edges and n vertices v1, v2,…, vn,
then
n
(vi) = 2m
i = 1
Theorem: In any graph, there are an even number of vertices
of odd degree.
Divide the vertices into two groups: even degree x1,x2,…,xm and
odd degree y1,y2,…,yk
S= (xp) (even)
T= (yj)
S+T = (Vi)= 2m (even)
thus T is even, therefore k is even
66. Euler - theorems
1. A connected graph G is Eulerian if and only if G is connected
and has no vertices of odd degree
e
a b
c d
f
Building a simple path:
{a,b}, {b,c}, {c,f}, {f,a}
Euler circuit constructed if all edges
are used. True here?
67. Euler - theorems
1. A connected graph G is Eulerian if and only if G is connected
and has no vertices of odd degree
c d
e
Delete the simple path:
{a,b}, {b,c}, {c,f}, {f,a}
C is the common vertex for this
sub-graph with its “parent”.
68. Euler - theorems
1. A connected graph G is Eulerian if and only if G is connected
and has no vertices of odd degree
c d
e
Constructed subgraph may not be connected.
C is the common vertex for this sub-graph
with its “parent”.
C has even degree.
Start at c and take a walk:
{c,d}, {d,e}, {e,c}
69. Euler - theorems
1. A connected graph G is Eulerian if and only if G is connected
and has no vertices of odd degree
e
a b
c d
f
Building a simple path:
{a,b}, {b,c}, {c,f}, {f,a}
Euler circuit constructed if all edges
are used. True here?
70. Euler - theorems
1. A connected graph G is Eulerian if and only if G is connected
and has no vertices of odd degree
c d
e
Delete the simple path:
{a,b}, {b,c}, {c,f}, {f,a}
C is the common vertex for this
sub-graph with its “parent”.
71. Euler - theorems
1. A connected graph G is Eulerian if and only if G is connected
and has no vertices of odd degree
c d
e
Constructed subgraph may not be connected.
C is the common vertex for this sub-graph
with its “parent”.
C has even degree.
Start at c and take a walk:
{c,d}, {d,e}, {e,c}
72. Euler - theorems
1. A connected graph G is Eulerian if and only if G is connected
and has no vertices of odd degree
a b
c d
e
f
“Splice” the circuits in the 2 graphs:
{a,b}, {b,c}, {c,f}, {f,a}
“+”
{c,d}, {d,e}, {e,c}
“=“
{a,b}, {b,c}, {c,d}, {d,e}, {e,c}, {c,f}
{f,a}
73. Representation- Incidence Matrix
e1 e2 e3
a 1 0 0
b 1 1 0
c 0 1 1
d 0 0 1
e 0 0 0
f 0 0 0
a b
c d
e
f
e1
e2
e3
e4
e5
e6
e7
e4 e5 e6 e7
0 0 0 1
0 0 0 0
0 1 1 0
1 0 0 0
1 1 0 0
0 0 1 1
74. Theorem?
If a graph G contains a cycle from v to v, G
contains a simple cycle from v to v?
A cycle (or circuit) is a path of nonzero
length from v to v with no repeated
edges.
A simple cycle is a cycle from v to v,
there are no repeated vertices.
75. Homework 4- Part one
Write a program to obtain Euler Circuits.
Include a simple user interface to “input” the graph.
Recommended: Input graph from file (XMLGraph
format).
Input graphs can be Eulerian, no need for checking
“non” Euler graphs
Minimum of 15 edges (no more than 20 edges
needed)
Simple documentation
Include a sample graph, if needed, to test
Any programming language
Due on final exam
76. Hamiltonian Graph
Hamiltonian path (also called traceable path) is a
path that visits each vertex exactly once.
A Hamiltonian cycle (also called Hamiltonian
circuit, vertex tour or graph cycle) is a cycle that
visits each vertex exactly once (except for the
starting vertex, which is visited once at the start and
once again at the end).
A graph that contains a Hamiltonian path is called a
traceable graph. A graph that contains a
Hamiltonian cycle is called a Hamiltonian graph.
Any Hamiltonian cycle can be converted to a
Hamiltonian path by removing one of its edges, but a
Hamiltonian path can be extended to Hamiltonian
cycle only if its endpoints are adjacent.
77. A graph of the vertices of a dodecahedron.
Is it Hamiltonian?
Yes
.
78. This one has a Hamiltonian path, but not a
Hamiltonian tour.
Hamiltonian Graph
80. Hamiltonian Graph
Similar notions may be defined for directed graphs, where
edges (arcs) of a path or a cycle are required to point in
the same direction, i.e., connected tail-to-head.
The Hamiltonian cycle problem or Hamiltonian circuit
problem in graph theory is to find a Hamiltonian cycle in a
given graph. The Hamiltonian path problem is to find a
Hamiltonian path in a given graph.
There is a simple relation between the two problems. The
Hamiltonian path problem for graph G is equivalent to the
Hamiltonian cycle problem in a graph H obtained from G by
adding a new vertex and connecting it to all vertices of G.
Both problems are NP-complete. However, certain classes
of graphs always contain Hamiltonian paths.
81. Hamiltonian Graph
DIRAC’S Theorem: if G is a simple graph with n
vertices with n ≥ 3 such that the degree of every
vertex in G is at least n/2 then G has a Hamilton
circuit.
ORE’S Theorem: if G is a simple graph with n
vertices with n ≥ 3 such that deg (u) + deg (v) ≥ n
fro every pair of nonadjacent vertices u and v in G,
then G has a Hamilton circuit.
82. Traveling salesperson problem
The problem of finding a Hamiltonian
cycle in a graph
Given a weighted graph G
Vertices: cities, Edge weights: distance
Find a minimum-length Hamiltonian cycle.
Find a shortest route in which the salesperson can
visit each city one time, starting and ending at the
same city.
84. Planar Graphs:
A graph is called planar if it can be drawn in the
plane such that its edges do not cross. Such a
drawing is called a planar representation of the
graph.
Is K4 planar?
85. Other examples:
Graph Q3
Planar representation
Of Q3
Question: Draw K3,3 .Is it planar?
If not, justify your answer. Look at p-605 (Rosen)
86. Euler’s Formula:
A planar representation of a graph splits the plane into
regions, including an unbounded region. The graph in
planar representation given by
splits the plane into six regions. Let G be a connected
planar simple graph with e edges and v vertices. Let r
be the number of regions in a planar representation of
G. Then v – e + r = 2.
R1
R2
R3
R4
R5
R6
87. Example:
Suppose that a connected planar simple graph has
20 vertices, each of degree 3. Into how many
regions does a representation of this planar graph
split the plane? Ans: 12
Corollary 1. If G is a connected planar graph with e
edges and v vertices where v >= 3, then e<= 3v -6.
Corollary 2. If G is a connected planar simple
graph, then G has a vertex of degree not exceeding
five.
88. Degree of a region:
Degree of a region is defined to be the number of
edges on the boundary of this region. When an edge
occurs twice on the boundary, it contributes two to
the degree.
R1
R2
R3
Degree of R1 = 3
Degree of R2 = 6
Degree of R3 = 7
Using previous corollary you can prove that K5 is
not a planar graph.
89. Kuratowski’s Theorem
(Characterization of planarity of graph with the
help of Homeomorphism)
Statement: A graph is nonplanar if and only if it
contains a subgraph homeomorphic to K3,3 or K5.
Hence G is nonplanar
90. Coloring of a graph:
Definition 1: A coloring of a simple graph is the
assignment of a color to each vertex of the graph
so that no two adjacent vertices are assigned the
same color.
Definition 2: The chromatic number of a graph is
the least number of colors needed for a coloring of
the graph and it is denoted by )
(G
92. Four Color Theorem:
The chromatic number of a planar graph is not
greater than four.
Non planar graphs can have arbitrarily large
chromatic number.
Consider two maps in the next slide and their
corresponding graph which are called dual graph of
the map, i.e. the problem of coloring the regions of
a map is equivalent to the problem of coloring the
vertices of the dual graph.
Note that in the left map, 4 colors are sufficient and
in the right map 3 colors are sufficient.
93.
94. Application of Graph Coloring:
(Welch Powell Algorithm):
How can the final exams at a university be scheduled
so that no student has two exams at the same time?
Modeling: Model the problem into a graph, with
vertices representing courses and with an edge
between two vertices if there is a common student in
the courses they represent
Each time slot for a final exam is represented by a
different color.
Thus a scheduling of the exams corresponds to a
coloring of the associated graph.
95. Continued:
Suppose that there are 7
finals to be scheduled.
Number of the courses
from 1 to 7. Suppose
following pairs of courses
have common students:
1 &2 , 1 & 3, 4 & 1, 1 &
7, 2 & 3, 2 & 4, 2 & 5, 2
& 7, and so on….
The graph associated is
given by:
97. Welch Powell Algorithm
Order the vertices of G according to
decreasing degrees. For above graph:
Vertices: 2 3 7 4 5 6 1
Degrees: 5 5 5 5 4 4 4
Colors: B G P P R B R
Example 8.21 (Look at graph on p-219
Schaum’s Series)
99. Application Examples
Useful for locating items in a list
Used in Huffman coding algorithm
Study games like checkers and chess to
determine winning strategies
Weighted trees used to model various
categories of problems
100. Trees
Definition: A connected undirected
graph with no simple circuits
Characteristics
- No multiple edges
- No loops
102. Which of the following graphs are
trees?
a
c
b
d
f
e
b
a
e
a
c
d
f
a
b c
d
e
a
b
d
c
e
f
A B C D
A & B : Trees
C: Not a Tree (Cycle abdcfa)
D: Not a Tree (Not connected). However it is a forest i.e. unconnected
graphs
103. Sub-trees
A subtree of a tree T is a tree T' such that
V(T') V(T) and
E(T') E(T)
H
L
K
J T
S
E
M
104. Tree Theorem -Theorem 1
An undirected Graph is a tree if and only if there
is a unique simple path between any two of its
vertices
Proof (=>) The graph is a Tree and it does not have cycles.
If there were cycles then there will be more than one simple
path between two vertices. Hence, there is a unique simple
path between every pair of vertices
Proof (<=) There exists a unique simple path between any
pair of vertices. Therefore the graph is connected and there
are no cycles in the graph. Hence, the graph is a tree
105. Tree terminology
Rooted tree: one vertex
designated as root and every
edge directed away from the
root
Parent: u is the parent of v
iff (if and only if) there is an
edge directed from u to v
Child: v is called the child of u
Every non-root node has a unique
parent (?)
106. Siblings: vertices with same parent
Ancestors: all vertices on path from the root to this
vertex, excluding the vertex
Descendants: Descendants of vertex v are all vertices
that have v as an ancestor
Internal Nodes: Nodes that have children
External or Leaf Nodes: Nodes that have no children
Tree terminology
107. A
B C
D E F
G H I J K
Root Node = A
Internal Nodes = A, B, C, E
and F
External Nodes or leaves = D,G, H
,I, J and K
Given : Tree rooted at A
Find: Descendants (B), Ancestor (A), Siblings (G) ?
108. Definition - Level
The level of vertex v in a rooted tree
is the length of the unique path from
the root to v
The level of root is 1
112. Ordered Rooted Trees:
An ordered rooted tree is a rooted tree where
the children of each internal vertex are shown in
order from left to right.
113. Contd:
Observe that number of decimal points in any label is
one less than the level of the vertex. We refere this
system of labeling as Universal address system for an
ordered rooted tree.
The addresses in the previous tree are linearly
ordered as follows:
0 , 1, 1.1, 1.2, 1.2.1, 1.2.2, 1.2.2.1, 2, 2.1, 3, 3.1,
3.1.1, 3.2, 3.2.1, 3.2.1.1, 3.2.2
This is also called Dictionary order or Lexicographical
order.
114. Definition: m-ary trees
Rooted tree where every vertex has no more than
‘m’ children
Full m-ary if every internal vertex has exactly ‘m’
children (i.e., except leaf/external vertices).
m=2 gives a binary tree
A complete m - ary tree is a full m-ary tree where
every leaf is at the same level.
Exercise: Construct a complete binary tree of
height 4 and a complete 3-ary tree of height 3.
116. Definition: Binary Tree
Every internal vertex has a maximum of 2
children
An ordered rooted tree is a rooted tree where
the children of each internal vertex are
ordered.
In an ordered binary tree, the two possible
children of a vertex are called the left child
and the right child, if they exist.
120. Theorem ????
Let T be a graph with n vertices.
The following are equivalent
T is a tree
T is connected and acyclic
T is a connected and has n-1 edges
T is acyclic and has n-1 edges
121. Tree Traversal
Information is stored in Ordered Rooted tree
Methods are needed for visiting each vertex of the
tree in order to access data
Three most commonly used traversal algorithms are:
1. Preorder Traversal
2. Inorder Traversal
3. Postorder Traversal
122. Preorder Traversal
Let T be an ordered rooted tree with root r
If T has only r, then r is the preorder traversal. Otherwise,
suppose T1, T2 are the left and right subtrees at r. The
preorder traversal begins by visiting r. Then traverses T1
in preorder, then traverses T2 in preorder.
124. Preorder Traversal
Procedure preorder (T: ordered rooted tree)
r := root of T
List r;
for each child c of r from left to right
T(c) := subtree with c as its root
preorder(T(c));
End.
125. Inorder Traversal
Let T be an ordered rooted tree with root r.
If T has only r, then r is the inorder traversal. Otherwise, suppose
T1, T2 are the left and right subtrees at r. The inorder traversal
begins by traversing T1 in inorder. Then visits r, then traverses T2
in inorder.
127. Inorder Traversal
Procedure inorder (T: ordered rooted tree)
r = root of T
if r is a leaf list r
Else
{
l = first child of r from left to right;
T(l) = subtree with l as its root
inorder (T(l));
list r;
for each child c of r except for l from left to right
{
T(c) = subtree with c as its root
inorder(T(c));
}
}
End.
128. Postorder Traversal
Let T be an ordered rooted tree with root r.
If T has only r, then r is the postorder traversal. Otherwise,
suppose T1, T2 are the left and right subtrees at r. The postorder
traversal begins by traversing T1 in postorder. Then traverses T2
in postorder, then ends by visiting r.
130. Postorder Traversal
Procedure postorder (T: ordered rooted tree)
r := root of T
for each child c of r from left to right
T(c) := subtree with c as its root
postorder(T(c));
End.
List r.
131. Infix, Prefix and Postfix
Notation
We can represent complicated expressions, such as
compound propositions, combinations of sets, and
arithmetic expressions using ordered rooted trees.
A binary tree representing arithmetic expression
+ -
x y x 4
+ 2
x
/
- 3
x 4
+
y
+ 2
y
x
/
- 3
x 4
132. Thus in this way, for such a given expression
we can draw an ordered rooted tree in which
the operations are the internal nodes and
variables are leaves.
# We obtain the infix form of an expression when
we traverse its rooted tree in inorder. In inorder
traversal it is necessary to include parentheses
whenever we encounter an operation because the
infix form of the expressions (x+y)/(x+3) ,
(x+(y/x))+3, and x+(y/(x+3)) will be x+y/x+3.
# The prefix form of an expression is obtained by
traversing its rooted tree in preorder. Expression
written in prefix form are said to be in
Polish Notation, which is named after the Polish
Logician Jan Lukasiewicz. No parentheses is needed.
133. # Sequentially the postfix form are said to be
reverse Polish Notation. (Parentheses not
needed)
What is the value of the prefix
expression + - * 2 3 5 / 2 3 4
Ans : 3
134. Given: Graph G=(V, E), |V|=n Cost function c: E R .
Goal: Find a minimum-cost spanning tree for V i.e., find
a subset of arcs E* E which connects any two nodes
of V with minimum possible cost
Example:
2
3
3
4
4
5
7
8
e
b
c
d
a
2
3
3
4
4
5
7
8
e
b
c
d
a
G=(V,E) Min. span. tree: G*=(V,E*)
Red bold arcs are in E*
Minimum Spanning Tree (Min SPT)
136. Prims Algorithm
Procedure Prims (G: weighted connected undirected graph with n
vertices)
T := a minimum-weight edge
For i := 1 to n-2
Begin
e:= an edge of minimum weight incident to a vertex in T and
not forming a simple circuit in T if added to T
T := T with e added
End {T is a minimum spanning tree of G}
142. Kruskal’s algorithm
Procedure Kruskal (G: weighted connected undirected graph
with n vertices)
T := empty graph
For i := 1 to n-1
Begin
e := any edge in G with smallest weight that does not form
simple circuit when added to T
T := T with e added
End { T is the minimum spanning tree of G}
148. Prim’s and Kruskal’s
Prim’s
Chosen edges: minimum weight & incident
to a vertex already in the tree.
Kruskal’s
Chosen edge: minimum weight & not
necessarily incident to a vertex in the tree.
Understand the difference:
http://students.ceid.upatras.gr/~papagel/project/prim.htm
http://students.ceid.upatras.gr/~papagel/project/kruskal.htm
