An introduction to graph theory - Indiana State Universitycs.indstate.edu/~arash/adslec8.pdf · An...

An introduction to graph theory

Arash Rafiey

13 October, 2015

Arash Rafiey An introduction to graph theory

Definition of Graph

Let V be a finite nonempty set and let E ⊆ V × V .

(V ,E ) is called a digraph where V is a set of vertices and E iscalled a set of (directed) edges or arcs.

When the order does not matter (relation is symmetric) we have agraph G = (V ,E ) and E (G ) ⊆ {{u, v}|u, v ∈ V (G )}

a

b c

d e

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

a

b c

de

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

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


Definition of Graph

Let V be a finite nonempty set and let E ⊆ V × V .

(V ,E ) is called a digraph where V is a set of vertices and E iscalled a set of (directed) edges or arcs.

When the order does not matter (relation is symmetric) we have agraph G = (V ,E ) and E (G ) ⊆ {{u, v}|u, v ∈ V (G )}

a

b c

d e

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

a

b c

de

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

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


For simplicity instead of edge {u, v} we write edge uv .

Two vertices u and v are called adjacent if uv is an edge of G .

We say v is a neighbor of u if uv is an edge of G .Let x , y be two vertices of graph G .

An x − y walk is an alternating sequence of vertices and edges,starting at x and ending at y .

If x = y then the walk is called closed.

A trail is a walk in which all the edges are distinct.

A path is a simple walk (no vertex repeated).

A cycle is a simple closed walk (no vertex repeated except thebeginning ).

































































Theorem

Let G = (V ,E ) be undirected graph with a, b ∈ V , a 6= b. If thereexists a trail from a to b, then there exists a path from a to b.

Proof.

Consider the shortens trail a, x1, x2, . . . , xn, b in G(ax1, x1x2, x2x3, . . . , xn−1xn, xnb are edges). If this trail is not apath then there exist k,m where xk = xm (x0 = a, xm+1 = b,k < m ). But then we can contract and get a shorter traila, x1, . . . , xk , xm+1, . . . , xn, b from a to b.


Theorem

Let G = (V ,E ) be undirected graph with a, b ∈ V , a 6= b. If thereexists a trail from a to b, then there exists a path from a to b.

Proof.

Consider the shortens trail a, x1, x2, . . . , xn, b in G(ax1, x1x2, x2x3, . . . , xn−1xn, xnb are edges). If this trail is not apath then there exist k,m where xk = xm (x0 = a, xm+1 = b,k < m ). But then we can contract and get a shorter traila, x1, . . . , xk , xm+1, . . . , xn, b from a to b.


Definition

A graph G is called connected if there is a path between any twodistinct vertices of G .

G H(connected) (not connected)

a

b c

d e

a

b c

d e

Definition

If G is not connected then it can be partitioned into pieces whereeach piece is a connected graph and is called a connectedcomponent .The number of connected components of G is denoted by κ(G ).


Example :

Let G = (V ,E ) be an undirected graph whose vertices are binaryn-sequences and the two vertices x , y are adjacent if they differ inexactly two positions.Find κ(G ).


Definition

A graph G is simple if there are no parallel edges and there is noself-loop.

Definition

If v is a vertex of graph G , then the degree of v , denoted deg(v)(dG (v), or dv ) is the number of edges incident to v . Is the numberof neighbors of v . The self-loop is counted twice.

If G is a simple graph and each vertex has degree k then G iscalled a k-regular graph.


Definition

A graph G is simple if there are no parallel edges and there is noself-loop.

Definition

If v is a vertex of graph G , then the degree of v , denoted deg(v)(dG (v), or dv ) is the number of edges incident to v . Is the numberof neighbors of v . The self-loop is counted twice.

If G is a simple graph and each vertex has degree k then G iscalled a k-regular graph.


Theorem (Handshaking)

For any graph G = (V ,E ) we have∑v∈V

deg(v) = 2|E (G )|.

Proof.

If an edge e has two end points u and v then it will contribute oneto each of deg(u), deg(v). If e is a self loop incident to vertex uthen it will contribute two to deg(u). In any case each edgecontributes two in the sum and identity follows.

Corollary

In any graph G the number of vertices of odd degree must be even.




deg(v) = 2|E (G )|.

Proof.


Corollary





deg(v) = 2|E (G )|.

Proof.


Corollary



Some families of Graphs

1) A graph G is called complete if every vertex is adjacent toevery other vertex.

2) For every n ≥ 2, the n-path-graph, denoted by Pn is just a pathon n vertices.

V (Pn) = {v1, v2, . . . , vn} and E (Pn) = {v1v2, v2v3, . . . , vn−1vn}.

3) For every n ≥ 2, the n-cycle-graph, denoted by Cn is just a cycleon n vertices.

V (Cn) = {v1, v2, . . . , vn}, E (Cn) = {v1v2, v2v3, . . . , vn−1vn, vnv1}.

4) For every n ≥ 2, the n-hypercube, denoted by Qn has vertex set

V (Qn) = {length n bit string} and two vertices are adjacent iftheir bit strings differ in exactly one position.





























Bipartite Graphs

Definition

A graph G = (V ,E ) is bipartite if V can be partitioned into U,Wsuch that each edge of G has one end point in U and one endpoint in W .

ExampleFor which n, Cn is a bipartite ?

For which value of n, Kn is bipartite ?

For which value of n, Pn is bipartite ?

For which value of n, Qn is bipartite ?


Bipartite Graphs

Definition







Bipartite Graphs

Definition







Bipartite Graphs

Definition







Bipartite Graphs

Definition







Definition

For positive integers n,m, the complete bipartite graph Kn,m hasthe following vertex and edge sets :

V (Kn,m) = {u1, u2, . . . , un} ∪ {v1, v2, . . . , vm}E (Kn,m) = {uivj |1 ≤ i ≤ n, 1 ≤ j ≤ m}

K3,3

K2,3


Degree Sequence

Give any graph, we can obtain the degree sequence (d1, d2, . . . , dn)of its vertices v1, v2, . . . , vn.

We are given a sequence (d1, d2, . . . , dn). Can we decide whetherthere exists a graph G whose degree sequence is (d1, d2, . . . , dn) ?

If this is the case then (d1, d2, . . . , dn) is called a graphic sequence.

(2, 2, 2) is graphic but (2, 3, 4, 3, 2, 3) is not graphic.


Degree Sequence






Degree Sequence






Theorem

Suppose π = (d1, d2, . . . , dn) is a sequence withn > d1 ≥ d2 ≥ · · · ≥ dn.

1 If π is graphic then there is a graph G withV (G ) = {v1, v2, . . . , vn} and deg(vi ) = di and the neighborsof v1 are v2, v3, . . . , vd1+1.

2 π is graphic if and only if(d2 − 1, d3 − 1, . . . , dd1+1 − 1, dd1+2, . . . , dn) is graphic.

Proof.

Proof of (1). Since π is graphic, there is a graph G with verticesv1, v2, . . . , vn and degree sequence π.We may assume that G = (V ,E ) is such a graph that N(v1) ∩ S ismaximum where S = {v2, v3, . . . , vd1+1} (N(v1) is theneighborhood of v1). If v1 is adjacent to all the elements in S thenwe are done. Otherwise there exists some vk such that v1vk 6∈ Eand hence there exits ` > d1 + 1 where v1v` ∈ E .


Proof.

Since k < `, vk has as many neighbors as v` and hence vk has aneighbor vj that is not neighbor of v`. Now create new graph G ′

by removing edge v1v` and adding edge v1vk and removing edgevkvj and adding edge vjv`. G ′ has the same degree sequence as Gbut N(v1) ∩ S increases in G ′, a contradiction.

Proof of (2). If π is graphic then by (1) there is G with verticesv1, v2, . . . , vn and degree sequence π where v1 is adjacent tov2, v3, . . . , vd1+1. If we remove v from G then we have a sequence(d2−1, d3−1, . . . , dd1+1−1, dd1+2, . . . , dn). Conversely, if we startwith graphic sequence (d2 − 1, d3 − 1, . . . , dd1+1 − 1, dd1+2, . . . , dn)associated with graph G ′ then we add a new vertex v and connectit to the vertices of G ′ with degrees d2 − 1, d3 − 1, . . . , dd1+1 − 1.This way we obtain G with degree sequence π.


Proof.

Since k < `, vk has as many neighbors as v` and hence vk has aneighbor vj that is not neighbor of v`. Now create new graph G ′

by removing edge v1v` and adding edge v1vk and removing edgevkvj and adding edge vjv`. G ′ has the same degree sequence as Gbut N(v1) ∩ S increases in G ′, a contradiction.

Proof of (2). If π is graphic then by (1) there is G with verticesv1, v2, . . . , vn and degree sequence π where v1 is adjacent tov2, v3, . . . , vd1+1. If we remove v from G then we have a sequence(d2−1, d3−1, . . . , dd1+1−1, dd1+2, . . . , dn). Conversely, if we startwith graphic sequence (d2 − 1, d3 − 1, . . . , dd1+1 − 1, dd1+2, . . . , dn)associated with graph G ′ then we add a new vertex v and connectit to the vertices of G ′ with degrees d2 − 1, d3 − 1, . . . , dd1+1 − 1.This way we obtain G with degree sequence π.


Algorithm to detect graphic sequence

Graphic (n > d1 ≥ d2 ≥ d3 ≥ · · · ≥ dn)1. while d1 > 0

2. Set (d ′1, d′2, . . . , d

′n−1) be a non-decreasing permutation of

(d2 − 1, d3 − 1, . . . , dd1+1 − 1, dd1+2, . . . , dn)

3. Set n = n − 1 and (d1, d2, . . . , dn) = (d ′1, d′2, . . . , d

′n)

4. if di < 0 then output NO exit

5. else if d1 = 0 then output YES exit


ExampleInitial Sequence : (4,4,3,3,2,2)

(1) : (3,2,2,1,2)

Sort : (3,2,2,2,1)

(2) : (1,1,1,1)

(3) : (0,1,1)

Sort (1,1,0)

(4) : (0,0) True

Is the sequence (5, 5, 3, 5, 3, 3, 3, 3) is graphic ?



(1) : (3,2,2,1,2)

Sort : (3,2,2,2,1)

(2) : (1,1,1,1)

(3) : (0,1,1)

Sort (1,1,0)

(4) : (0,0) True




(1) : (3,2,2,1,2)

Sort : (3,2,2,2,1)

(2) : (1,1,1,1)

(3) : (0,1,1)

Sort (1,1,0)

(4) : (0,0) True




(1) : (3,2,2,1,2)

Sort : (3,2,2,2,1)

(2) : (1,1,1,1)

(3) : (0,1,1)

Sort (1,1,0)

(4) : (0,0) True




(1) : (3,2,2,1,2)

Sort : (3,2,2,2,1)

(2) : (1,1,1,1)

(3) : (0,1,1)

Sort (1,1,0)

(4) : (0,0) True




(1) : (3,2,2,1,2)

Sort : (3,2,2,2,1)

(2) : (1,1,1,1)

(3) : (0,1,1)

Sort (1,1,0)

(4) : (0,0) True




(1) : (3,2,2,1,2)

Sort : (3,2,2,2,1)

(2) : (1,1,1,1)

(3) : (0,1,1)

Sort (1,1,0)

(4) : (0,0) True




(1) : (3,2,2,1,2)

Sort : (3,2,2,2,1)

(2) : (1,1,1,1)

(3) : (0,1,1)

Sort (1,1,0)

(4) : (0,0) True



Subgraphs

Definition

Let G = (V ,E ) be a graph. Graph H ′ = (V ′,E ′) is a subgraph ofG if V ′ ⊆ V and E ′ ⊆ E .

C5 is a subgraph of K6, and K1,K2, . . . ,K5 are all subgraph of K6.


Subgraphs

Definition

Let G = (V ,E ) be a graph. Graph H ′ = (V ′,E ′) is a subgraph ofG if V ′ ⊆ V and E ′ ⊆ E .

C5 is a subgraph of K6, and K1,K2, . . . ,K5 are all subgraph of K6.


Graph Isomorphism

Definition

Let G = (V ,E ), G ′ = (V ′,E ′) two graphs. Suppose f : V → V ′ isa one-to-one function.

1 f preserve adjacency if for every uv ∈ E , f (u)f (v) ∈ E ′.

2 f preserve non-adjacency if for every non adjacent verticesu, v then f (u), f (v) are non-adjacent.

3 f is a graph isomorphism from G to G ′ if it is bijective andpreserve both adjacency and non-adjacency. In this case wewrite G ∼= G ′.

a b c d

G G’

f(a)=c and f(b)=d f is isomorphism


Graph Isomorphism

Definition





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Graph Isomorphism

Definition





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G G’



Graph Isomorphism

Definition





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G G’



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