1 CS104 : Discrete Structures Chapter V Graph Theory

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CS104 : Discrete Structures

Chapter VGraph Theory

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A Graph, G = (V, E), is a data structure which consists of a nonempty set of vertices (or nodes), V, and a set of edges, E, that connect (some of) them.

The set of vertices V of a graph G may be infinite. A graph with infinite vertex set is called infinite graph. A graph with finite vertex set is called finite graph. For example,

V = {1, 2, 3, 4, 5} E = { (1,2), (1,3), (1,4), (2,3), (3,5), (4,5) }

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Vertex(Node)Edge

What is a Graph?

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Computer Networks

Electrical Circuits

Road Map

Computer

Resistor/Inductor/…

City

Application of Graphs

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An Undirected Graph is a graph where the edges have no directions.

The edges in an undirected graph are called Undirected Edges

For example, G = (V, E), where V = {1, 2, 3, 4, 5} and E = {(1,2), (1,3), (1,4), (2,3), (3,5),

(4,5)} A Directed Graph or Digraph is a graph where each edge has a

direction. The edges in a digraph

are called Arcs or Directed Edges For example, G = (V, E), where V = {1, 2, 3, 4, 5, 6} and

E = {(1,4), (2,1), (2,3), (3,2), (4,3), (4,5), (4,6), (5,3), (6,1), (6,5)}

The edge (1, 4) = 1→4 where 1 is the tail and 4 is the head

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Graph Categorization

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A graph with both directed and undirected edges is called a mixed graph.

Adjacent vertices: If (u, v) is an edge of the graph, then the nodes u and v are adjacent.

An edge (u, v) is incident to vertices u and v. For example, Vertices 1 and 2 are adjacent,

but vertices 2 and 5 are not adjacent A Weighted Graph is a graph where

all the edges are assigned weights. If the same pair of vertices have more

than one edges (multiple edges), that graph is called a Multigraph

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Loop or self edges: An edge (u, u) is called a self edge or a loop.

( 1,1 ) and ( 4,4 ) are self edges Graphs that may include loops,

and possibly multiple edges connecting same pair of nodes, are called pseudographs.

In simple graphs (or graphs) loops and multiple edges are not permitted

The complete graph on n vertices, Kn, is the simple graph that contains exactly one edge between each pair of distinct nodes.

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The cycle Cn, n≥3, consists of n vertices v1, v2, ….., vn, and edges (v1, v2), (v2, v3), … , (vn-1, vn), and (vn, v1).

That is, Cn consists of n vertices and n edges.

A simple graph G is called bipartite if its vertex set 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 in V2. When this condition holds, we call the pair (V1, V2) a bipartition of the vertex set V of G.

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Example 1:- C6 is bipartite, because

its vertex set can be partitionedinto two disjoint sets V1={v1,v3,v5}

and V2={v2,v4,v6}, and every edge

of C6 connects a vertex in V1 and a vertex in V2.

Example 2:- K3 is not bipartite, because if we divide the vertex set into two disjoint sets, one set must contain two vertices. If the graph were bipartite, these two vertices could not be connected by an edge, but in K3 each vertex is connected to every other vertex by an edge.

Theorem:- A simple graph is bipartite if and only if it is possible to assigns one of two different colors to each vertex of the graph so that no two adjacent vertices are assigned the same color.


V1

v1 .v3 .v5 .

V2

. v2

. v4

. v6

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A Subgraph of graph G=(V,E) is a graph H=(U,F) such that U⊆ V and F ⊆ E. A subgraph H is a proper subgraph of G if H≠G.

The union of two simple graphs G1=(V1, E1) and G2=(V2, E2) is the simple graph with vertex set (V1υV2) and edge set (E1υE2). The union of G1 and G2 is denoted by G1υG2. For example,

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G=(V,E)

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H=(U,F)


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G1

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G2

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G1 U G2

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Degree of a Vertex : In an undirected graph, the no. of edges incident to the vertex. Degree of vertex v is denoted by deg(v).

For example, deg(1)=3, deg(2)=2, deg(3)=2, deg(4)=1 and deg(5)=0.

A vertex of degree zero is called isolated vertex. A vertex of degree one is called pendant vertex. For example, vertex 4 is pendant and vertex 5 is isolated. The Handshaking Theorem: - Let G=(V, E) be an

undirected graph. Then 2|E| = ∑deg(v), ∀v ∈ V Example 3:- How many edges are there in a graph with 10

vertices each of degree six? Solution:- Because the sum of the degrees of the vertices

is 6x10=60, it follows that 2|E|=60. Therefore, |E|=30.

Graph Terminology

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In diagram, there are two types of degree of a vertex - in-degree and out-degree

In-degree: In-degree of a vertex v, denoted by deg-(v), is the no. of edges entering the vertex in a digraph

Out-Degree: Out-degree of a vertex v, denoted by deg+(v), is the no. of edges leaving the vertex

For example, deg-(1)=3 and deg+(1)=1.deg-(2)=1 and deg+(2)=1.

Theorem:- For a digraph G=(V, E), ∑deg-(v)+∑deg+(v)=|E|, ∀v∈V

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Graph Terminology

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Adjacency Matrix:- The Adjacency Matrix A=(ai,j) of a graph G=(V,E) with n nodes is an nXn zero-one matrix. Element of A can be defined as follows:

aij = 1, if (vi, vj) Є E,

= 0, otherwise Example 4: Find the adjacency matrices of the following

graphs.

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12345

101110

210100

311001

410001

500110

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1010

2001

3110

Representation of Graphs

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Adjacency Matrix of a Weighted Graph:- The weight of the edge can be shown in the matrix when the vertices are adjacent A nil value (0 or ∞) depending on the problem is used

when they are not adjacent Example 5: Adjacency matrix

to find the minimum distance between nodes

Example 6: Draw graph with the adjacency matrix

123

1∞4∞

2∞∞5

395∞

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4 5


123

1∞19

21∞∞

3∞8∞

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1 8

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Adjacency List:- An Adjacency list is an array of lists, each list showing the vertices a given vertex is adjacent to….

Example 7: Find the adjacency list of the following graph.

Adjacency List of a Weighted Graph:- The weight is included in the list

Example 8: Find the adjacency list of the following graph.

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12345

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Incidence Matrix:- Let G=(V,E) be a graph with v1, v2, …, vn are vertices and e1, e2, …., vm are edges. Then the incidence matrix with respect to this ordering of V and E is the nxm matrix M=[mij], where

mij = 1, when edge ej is incidence on vi,

= 0, otherwise Example 9: Find the incidence matrices of the following

graphs. e1e2e3e4e5e6

v1111000

v2100100

v3010110

v4001001

v5000011

e1e2e3

v1100

v2011

v3000


v1

v2

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v5

e2

e5

e1e4

e6

e3

v1

v2

v3

e1

e2 e3

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Def:- The simple graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic if there is one-to-one and onto function f from V1 to V2 with the property that a and b are adjacent in G1 if and only if f(a) and f(b) are adjacent in G2, for all a, b in V1. Such a function f is called an isomorphism.

Example 10: Show that the graphs G=(V, E) and H=(U, F) are isomorphic.

Proof:- The function f with f(v1)=u1, f(v2)=u4, f(v3)=u3, and f(v4)=u2 is a one-to-one correspondence between V and U. Next, to verify that this function preserves adjacency also. Adjacency vertices in G are v1 and v2, v1 and v3, v2 and v4, and v3 and v4. Also, the pairs f(v1)=u1 and f(v2)=u4, f(v1)=u1 and f(v3)=u3, f(v2)=u4 and f(v4)=u2, f(v3)=u3, and f(v4)=u2 are adjacent in H.

Isomorphism of Graphs

v1 v2

v3 v4

u1 u2

u3 u4

GH

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A a path a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along the edges of the graph.

There can be more than one path between two vertices. Vertex u is reachable from v if there is a path from u to v. Paths from 5 to 1 are 5, 3, 1 and 5, 3, 2, 1 A Simple Path is a path, where all the

vertices are distinct. For example, 1,4,5,3 is a simple path, but 1,4,5,4 is not Length of a path is the no. of edges considered in the path. Length of the path 1,4,5,3 is 3 A circuit is a path whose first and last vertices are the same. For example, the path 3,2,1,4,5,3 is a circuit

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Connectivity

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A Cycle is a circuit where all the vertices are distinct except for the first (and the last) vertex.

For example, 1,4,5,3,1 is a cycle, but 1,4,5,4,1 is not a cycle.

Hamiltonian Cycle: A Cycle that contains all the vertices of the graph.

1,4,5,3,2,1 is a Hamiltonian Cycle. A graph is said to be connected if there is at least one

path from every vertex to every other vertex in the graph. An Euler circuit in a graph G is a simple circuit containing

every edge of G. An Euler path in G is a simple path containing every edge of G.

For example, G has an Eulercircuit, but H does not have

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Euler Paths

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G

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Tree: A connected undirected graph that contains no cycles

Forest: A graph that does not contain a cycle

The Spanning Tree of a Graph G is a subgraph of G that is a tree and contains all the vertices of G.

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Tree

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Forest

Spanning Tree

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Graph

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Spanning Tree

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Tree

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End of Chapter V

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1 CS104 : Discrete Structures Chapter V Graph Theory