An Introduction to Graph Theory

7/29/2019 An Introduction to Graph Theory

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An Introduction to Graph Theory

Course 2

Crina Grosan


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The Konigsberg Bridge Problem


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Undirected graph Directed graph

isolated vertex

adjacent

loop

multipleedges

simple graph: an undirected graph without loop or multiple edges

degree of a vertex: number of edges connected (indegree, outdegree)

G=(V,E)

For simple graphs, deg(v Eiv Vi

) | | = 2

Definitions and Examples


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x y

path: no vertex can be repeateda-b-c-d-e

trail: no edge can be repeata-b-c-d-e-b-d

walk: no restrictiona-b-d-a-b-c

closed if x = yclosed trail: circuit (a-b-c-d-b-e-d-a,

one draw without lifting pen)closed path: cycle (a-b-c-d-a)

a

b

c

d

e

length: number of edges inthis (path,trail,walk)



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.tofrompatha

isethen ther,tofromtrailaexiststhereIf.,,

withgraph,undirectedanbe),(=LetTheorem.

ba

babaVba

EVG

a x b

remove any cycle on the repeatedvertices

Definition. Let G=(V,E) be an undirected graph. We call Gconnectedif there is a path between any two distinct vertices ofG.

a

b

c

d

e a

b

c

d

e

disconnected with

two components



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multigraph ofmultiplicity3

multigraphs



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Definitions and Representation

An undirected graph and its adjacency matrix representation.

An undirected graph and its adjacency list representation.


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.inesith verticincident wisin

ofedgeeachwhere,andifofsubgrapha

calledis),(thengraph,ais),(=If.Definition

11

11

111

VE

EEVVG

EVGEVG

=

a

b

cd

e

a

b

c

d

e

b

c

d

ea

cd

spanning subgraph

V1=V

Subgraphs, Complements, and Graph Isomorphism


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Definition: complete graph: Kn

a

b

c

d

e

K5

Definition: complement of a graph

G Ga

b

c

d

e

a

b

c

d

e



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Graph Isomorphism

1 2

3 4

a b

c

d

w x y z

.Ginadjacentare(y)and(x)ifonlyandifGinadjacentareGofyandxverticesany twoifmisomorphis

graphacalledis:functionAgraphs.undirected

twobe),(and),(Let.Definition

2

1

21

222111

VVf

EVGEVG

==



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Example q r

w

z xy

u t

v

a

b

cd

ef

g

h

i

j

a-q c-u e-r g-x i-z b-v d-y f-w h-t j-s, isomorphic

Example


a

b c

d

ef

1

23

4

56


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Regular graphs

A graph is said to be regular of degree if all local

degrees are the same number . A 0-regular graph isan empty graph, a 1-regular graph consists of

disconnected edges, and a 2-regular graph consists of

disconnected cycles. The first interesting case is therefore

3-regular graphs, which are called cubic graphs.


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For simple graphs, deg(v Eiv Vi ) | |

= 2Theorem.

Corollary . The number of vertices of odd degree must be even.

Example: a regulargraph: each vertex has the same degree

Is it possible to have a 4-regular graph with 10 edges?

2|E| = 4|V| = 20, |V|=5 possible (K5)

with 15 edges?

2|E| = 4|V| = 30

Vertex Degree: Euler Trails and Circuits

not possible!!!


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a

Find a way to walk about the city so as to cross each bridge exactlyonce and then return to the starting point.

The Seven Bridge of Konigsberg


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Definition. Let G=(V,E) be an undirected graph or multigraph with noisolated vertices. Then Gis said to have an Euler circuitif there is a circuitin G that traverses every edge of the graph exactly once. If there is anopen trail from ato bin Gand this trail traverses each edge in Gexactlyonce, the trail is called an Euler trail.

Theorem. Let G=(V,E) be an undirected graph or multigraph with noisolated vertices. Then Ghas an Euler circuit if and only ifGis connected

and every vertex in G has even degree.

All degrees are odd. Hence no Euler circuitfor the Konigsberg bridges problem !



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Corollary. An Euler trail exists in Gif and only ifGis connected and has exactlytwo vertices of odd degree.

two odd degree vertices

a b

add an edge

Theorem. A directed Euler circuit exists in Gif and only ifGis connectedand in-degree(v)=out-degree(v) for all vertices v.

two in, two out

Can you think of an algorithm to construct an Euler circuit?



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ExerciseEuler's House. Baby Euler has just learned to walk. He is curious to know if he can

walk through every doorway in his house exactly once, and return to the room he

started in. Will baby Euler succeed? Can baby Euler walk through every door exactly

once and return to a different place than where he started? What if the front door is

closed?


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Definition. A graph (or multigraph) Gis calledplanarifGcan be drawn in theplane with its edges intersecting only at vertices ofG.Such a drawing ofGis called an embeddingofGin the plane.

Example: K1,K2,K3,K4 are planar, Kn for n>4 are nonplanar.

K4 K5

applications: VLSI routing, plumbing,...

Planar Graphs


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bipartite graph complete bipartite graphs (Km,n)

K4,4

K3,3 is not planar.

Planar Graphs


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elementary subdivision (homeomorphic operation)

u w u v w

Definition. G1 and G2 are called homeomorphicif they are isomorphic or ifthey can both be obtained from the same loop-free undirected graph Hby a

sequence of elementary subdivisions.

a b

cde

a b

cde

a b

cde

a b

cde

Two homeomorphic graphs are simultaneously planar or nonplanar.

Planar graphs


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Degree of a region (deg(R)): the number of edges traversed in a shortestclosed walk about the boundary ofR.

R1

R2

R3

R4R5

R6

R7

R8

two different embeddings

deg(R1)=5,deg(R2)=3

deg(R3)=3,deg(R4)=7

deg(R5)=4,deg(R6)=3

deg(R7)=5,deg(R8)=6

deg( ) deg( ) | |R R Eii

ii= =

= = = =1

4

5

818 2 9 2

abghgfda

a b

c

d fg h

Planar Graphs


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a b

c

d

ef

g

1 2

3

4

5

6

1

6 5

4

2

3

An edge in Gcorresponds with an edge in Gd.

It is possible to have isomorphic graphs with respective duals thatare not isomorphic.

A dual graph of a planar graph


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Cut-set: a subset of edges whose removal increase the number ofcomponents

Example

a

b

c

d

e

f

g

h

cut-sets: {(a,b),(a,c)},{(b,d),(c,d)},{(d,f)},...

a bridge

For planar graphs, cycles in one graph correspond to cut-setsin a dual graphs and vice versa.

Cut set


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a path or cycle that contain every vertex

Unlike Euler circuit, there is no knownnecessary and sufficient condition for agraph to be Hamiltonian.

Examplea b c

d e f

g h

i

There is a Hamilton path, but no

Hamilton cycle.

an NP-complete problem

Hamilton Paths and Cycles


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Examplex

yy y y

x x

x

y

y

start labeling from here

4x's and 6y's, since xand ymustinterleave in a Hamilton path (or cycle),the graph is not Hamiltonian

The method works only for bipartite graphs.

The Hamilton path problem is still NP-complete when restrictedto bipartite graphs.



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Exercise. 17 students sit at a circular table, how many sittings arethere such that one has two different neighbors each time?

Consider K17, a Hamilton cycle in K17 corresponds to a seatingarrangements. Each cycle has 17 edges, so we can have(1/17)17(17-1)/2=8 different sittings.

12

3

4

5

6

17

16

15

1,2,3,4,5,6,...,17,1

12

3

4

5

6

17

16

15

1,3,5,2,7,4,...,17,14,16,1

12

3

4

5

6

17

16

15

1,5,7,3,9,2,...,16,12,14,1

14



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path.Hamiltondirectedacontainsalways)

a(calledgraphaSuch.inis),(or),(edgestheofone

exactlyvertices,of,pairdistinctanyforandverticeshas

i.e.,graph,directedcompleteabeLetTheorem.

*

**

tournament

Kxyyx

yxn

KK

n

nn



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cycle.Hamiltonacontainscontainsthen,,

tnonadjacenallfor)(deg+)deg(If3.|=|with

graphundirectedfree-loopabe),(=LetTheorem.

path.Hamiltonahas

graphthen thevertices,allfor2

1-

)deg(IfCorollary.

GVyx

nyxnV

EVG

nv



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A related problem: the traveling salesman problem

a

b

c

d

e3

4

13

5 4

32 Find a Hamilton cycle of shortest total distance.

2

For example, a-b-e-c-d-a with total cost=

1+3+4+2+2=12.



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Definition. IfG=(V,E) is an undirected graph, a proper coloring ofGoccurs whenwe color the vertices ofGso that if (a,b) is an edge in G, then aand bare coloredwith different colors. The minimum number of colors needed to properly color Giscalled the chromatic number ofG.

b

c

d

e

a3 colors are needed.

a: Redb: Green

c: Red

d: Blue

e: Red

In general, it's a very difficult problem (NP-complete).

(Kn)=n

(bipartite graph)=2

Graph Coloring and Chromatic Polynomials


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A related problem: color the map where two regions arecolored with different colors if they have same boundaries.

G

R e

B

B

R

Y

Four colors are enough for any map. Remaina mystery for a century. Proved with the aid

of computer analysis in 1976.

a b

c

d

f

a

b

c

d

e

f



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P(G,): the chromatic polynomial ofG=the number of ways

to color Gwith colors.

Example

(a) G=n isolated points, P(G,)=n.

(b) G=Kn, P(G,)=(-1)(-2)...(-n+1)=(n)

(c) G=a path ofn vertices, P(G,)=(-1)n-1.

(d) IfGis made up of components G1, G

2, ..., G

k,

then P(G,)=P(G1,)P(G2,)...P(Gk,).

Example

e

G eG eG'

coalescing the vertices


Documents

An Introduction to Graph Theory