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ECE-256alecture 2

256a 1

Malgorzata Marek-Sadowska

Electrical and Computer Engineering Department

Engineering I, room 4111

[email protected].

Reference materials

• C.L.Liu, “Introduction to Combinatorial Mathematics”, McGraw-Hill (1968)

• J.L.Gross and J.Yellen, “Graph Theory and its Applications,” Second Edition (DiscreteMathematics and Its Applications), Chapman & Hall/CRC, 2005.

• F. Buckley and M. Lewinter, “ A Friendly Introduction to Graph Theory, PrenticeHall”, 2002.

• J.M.Aldous, R.J.Wilson, “Graphs and Applications: An Introductory Approach” (withCD-ROM), Springer; 1st ed. 2000. Corr. 3rd printing edition (February 10, 2003)

256a 2

– ary hartrand, Introductory raph Theory, over ublications; Unabridged edition(December 1, 1984)

– R. J. Trudeau, Introduction to Graph Theory (Dover Books on Advanced Mathematics)Dover Publications; 2Rev Ed edition (February 9, 1994)

• And many more books on combinatorial mathematics and graph theory

Graph Theory

• Graph G(V,E) consists of 2 sets:

V= finite set of vertices {v}

E=set of edges {e}

256a 3

• n rec e grap

Each edge e ε E associated with an unordered pair

of vertices:

u e ve = (u,v)

Vertex adjacency: vertices u,v, are

adjacent if e=(u,v) ε E.

Sim le ra h: ra h w ith n o arallel e d es o r self l oo s

Some terminology

Incidence: edge is incident on its vertices.

A

B

C

D

e1 e2

e3

256a 4

.

Degree of vertex: d(v) = number of edges incident to v.

Path: sequence of edges {ei, ei+1, …}

Euler path: path which contains each edge once and only

once.

Path length: number of edges in the path.

B

Ce1 e2

E

F

A

B

C

D

e1 e2

e3

E

F

G1

e4

e5e6

e7

e8e9

In G1: {e1,e4,e9,e7,e5} –path of length 5.

Circuit – closed path.

Acyclic graph: graph with no cycles (ex. G2).

256a 5

A De3

G2

Euler circuit: circuit which contains each edge once and only once.

Example:{e10,e4,e1,e2,e9,e8,e7,

e6, e3, e12, e5,e11}

A

B

C

D

e1 e2

e3

E

F

G3

e4

e5

e7

e8e9

e10

e6e11

e12

A

B

C

D

E

F

G4

Connected graph: if for every u,v ε V there exists

A path from u to v.

Subgraph G’ of G:

G’(V’,E’): V’ V, E’ E, such that an edge {u,v} ε E’,

256a 6

only if (u,v) ε E. (G5 is a subgraph of G4)

Proper subgraph: E’ E, V’ V.

A

B

D

F

G5( G5 is also a proper subgraph of G4.



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A

B

C

D

E

F

Spanning subgraph: subgraph with V=V’.

G6Component of G: maximal

connected subgraph of G. c2

256a 7

A

B

D

FC

I

H

c1

Maximal subgraph G’ of G w.r.t. some property P: if G’

has the property P, and G’ is not a proper subgraph of any

other subgraph of G having the property P.

Component of G: maximal

connected subgraph of G. c2

256a 8

A

B

D

FC

I

H

c1

A

B

C

D

E

F

Minimal subgraph G’ of G w.r.t. some

property: G’ has the property P, and

no subgraph of G having the property

P is a proper subgraph of G’.

GExample: The subgraph induced by the

red edges in G is a minimal spanning

subgraph of G.

256a 9

A

B

C

D

E

F

G1

Example: The subgraph induced by the

red edges in G1 is a minimal connected

spanning subgraph of G1.

Directed graph (digraph): D(V,E)

E=directed edges or arcs; e= (u,v) is an ordered pair of

vertices.

u v

Directed path: sequence of arcs { (u,v), (v,w)….}

256a 10

Directed circuit (cycle): closed directed path.

Acyclic digraph: digraph with no directed circuits.

In-degree:

}),(|{)();( E vwwvvd ∈=Γ −−

Out-degree:

}),(|{)();( E wvwvvd ∈=Γ ++

v

v

256a 11

For every graph:

)()(||

1

||

1

=

+

=

−=

v

iii

v

i

vd vd

Strongly connected:if there is a directed path u->v

and v->u for all vertices in D.

(All vertices are strongly connected)

256a 12



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Strongly connected component:

maximal strongly connected subgraph.

Minimally connected:

IfD is stron l connected and removal

256a 13

of any edge destroys the “strongly

connected” property.

Cutset of a connected graph:

Minimal set of edges S in G whose removal

From G disconnects G, I.e. G-S is disconnected,

And has exactly 2 components.

256a 14

C1

C2

Cutset of minimum cardinality = min cutset.

Cut: set of edges having one end vertex in V1, and

the other in V2, and

V1 V2 = 0, V1 V2 = V∩ ∪

Min-cut = cut of minimum cardinality

v1 v2

256a 15

v3

v4 v5

v1 v2

v3

v4 v5

s s a cu , u no a cu-se

(not a minimal set of edges).

Resultant graph has 3 connected components,

not 2.

The first graph

The bridges of Konigsberg

A

C

DA

C

D

256a 16

B

Problem: Start at any of the four land areas, walk through each bridge

exactly once, and return to the starting point ( Euler circuit).

Special graphs

Complete graph2

)1(||

−=

nn E hasK

n

K6

Clique of G: set of vertices , such that

the induced subgraph of G on S is a complete

V S ⊆

256a 17

graph.

Clique problem: determine if a given

simple graph has a clique of size k.

Regular graph: k-regular, if all have d(v)=k.V v∈

5-regular, also K6

Note: Kn is (n-1) regular.

Bipartite graph: graph whose vertices can be

256a 18

vertices in the same set.



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Complete bipartite graph: km,n

if |V1|=m, |V2|=n and there is an edge between every

pair if vertices , 11V v ∈

22V v ∈

256a 19

V1 V2 V1 V2

Complete bipartite

graph.

Bipartite graph.

K-partite graph: vertices V partitioned into k

subsets: s.t. each edge has one

vertex in some and the other in .k

V V V ,,21

iV j

V

3-partite graph.

256a 20

Isomorphism

Graphs represent structural relations between

vertices.

Two graphs G and H are isomorphic, if there is

- -

256a 21

of vertices and their sets of edges, such that the

corresponding edges of G,H are incident on the

corresponding vertices of G,H,

Basic graph theorems

The sum of the degrees = 2 x the number of edges.

The number of vertices of odd degree is even.

E vd n

ii

2)(1

=−

256a 22

A graph G has an Euler path if and only if it is connected

and has two or no vertices of odd degree.

A graph G has an Euler circuit if and only if it is connected

and all vertices or of even degree.

In an acyclic digraph D there is at least one vertex

u with and at least one vertex v with0)( =ud in

0)( =ud out

Directed Euler circuit

256a 23

A directed graph contains an Euler

circuit if and only if its underlying graph is

connected and, for all vertices v :

)()( vd vd inout =

1)()( += ad ad inout

Directed Euler Path

A directed graph contains an Euler path, iff its

underlying graph is connected and contains one

vertex , a, such that

and at most one vertex, b, such that

256a 24

1)()( += bd bd out in

while for all other vertices, v

)()( vd vd inout

=



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Hamiltonian Circuit: a circuit in which

every vertex appears once and only once.

Dodecahedron

(Hamilton 1857)

256a 25

Hamiltonian path: a path in which every vertex

appears once and only once. No efficient

techniques exist for finding Hamiltonian

circuits in a graph.

Related problem:

Traveling Salesman Problem

Visit all cities exactly once and return to starting pointtraveling minimum distance.

256a 26

F n Ham ton an c rcu t o m n mum engt n a g ven

weighted graph.

Vertex: city to be visited

Edge: road between 2 cities

W(e): length of the road.

)(min: ewTSP

A necessary condition that a bipatrite graph has

a Hamiltonian circuit is that it has the same

number of vertices in each set.

A complete (directed) graph is one which has a

(directed) edge between every pair of vertices.

256a 27

Every complete directed graph has a Hamiltonian

path.

Trees

A tree is a connected graph with no circuits.

256a 28

A spanning tree of a graph is a tree which

contains all the vertices of the graph.

Minimum spanning tree theorem

If

• the vertices of a graph, G, are divided into

two disjoint sets, A and B; and

• e is the minimum length edge with one vertex

in A and another in B

256a 29

then there exists a minimum spanning tree,

T, which contains e.

A Be e is the min length edge.

Minimum spanning tree

Prim’s algorithm

1. Draw in shortest edge

2. Thereafter, draw in shortest edge which does

not complete a circuit.

256a 30

2

10

125

8

3

10

2

10

125

8

3

10



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Shortest path in a directed graph

4

3

2

6

7 2 32

7

4S T

256a 31

2 26

6

35

Dijkstra’s Algorithm

1. Set L(S) = 0, rest L(v) =

2. Find unmarked vertex u such that L(u) = min.

If u = T, STOP.

3. For each e = (u,v), set

L(v)=min(L(v), L(u) + l(e))4. Mark u, go to 2.

∞

256a 32

4

3

2

2

6

7 2 32

7

26

6

35

4

S T

(2)

(3)

(4)

(5)

(6)

(7)

(8)


4

3

2

6

7 2 32

7

4

∞

∞∞

0

S T

256a 33

2 26

6

35

∞ ∞


4

3

2

6

7 2 32

7

4

∞

∞

0

4

3

256a 34

2 26

6

35

∞2


4

3

2

6

7 2 32

7

4

∞

∞

0

4

3

7

256a 35

2 26

6

35

28


4

3

2

6

7 2 32

7

4

∞

∞

0

4

3

7

256a 36

2 26

6

35

28



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4

3

2

6

7 2 32

7

4

∞0

4

3

7

10

256a 37

2 26

6

35

28 7


4

3

2

6

7 2 32

7

4

∞0

4

3

7

10

6

256a 38

2 26

6

35

28 7


4

3

2

6

7 2 32

7

4

∞0

4

3

7

10

6

256a 39

2 26

6

35

28 7


4

3

2

6

7 2 32

7

4

0

4

3

7

10

6

13

9

256a 40

2 26

6

35

28 7


4

3

2

6

7 2 32

7

4

0

4

3

7

10

13

9

256a 41

2 26

6

35

28 7


4

3

2

6

7 2 32

7

4

0

4

3

7

10

13

9

256a 42

2 26

6

35

28 7

11



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4

3

2

6

7 2 32

7

4

0

4

3

7

10

613

9

256a 43

2 26

6

35

28 7

11

4 2

6

7 2 32

7

Dijkstra’s Algorithm

S T(3)

(4) (7)

256a 44

2 26

6

35

4

(2) (6)

(8)

Critical Path Problem

In scheduling problem (PERT)

• V = events, E = tasks

• l(e) = time needed to complete task e,

• the shortest possible completion time is determined

by the longest (critical) path.

256a 45

2

3

6

10

73

4

Find the longest path in directed graph.

Critical path algorithm

1. Set L(v) –0, u = S

2. For each e = (u,v), set

L(v) = max(L(v), L(u)+l(e))

3. Repeat 2 for all visited vertices.

256a 46

2

3

6

10

73

4

(2)

(3) (12)

(9)

(16)S

T

Independent Sets

A set of vertices of any graph is an independent set if

and only if no vertices in the set are adjacent.

B C B C

256a 47

A

G

EF D A

EF D

HB CH G

A

G

EF D

H

An independent set is maximal if the addition

of any other vertex causes it to become

dependent.

The independence number, I(G) of a graph, G,

256a 48

Independent set.



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C

A

EF D

An independent set is a dominating set if and

only if it is maximal.

B

256a 49

G H

Non-max independent sets: EH, BFG,AC.

Max independent sets: ACH, BDFG.

In any graph G: I(G)>=D(G).

0

0

0

0

256a 50

0

0

0

A maximal independent set on the chessboard

The covering problem

Given a binary matrix, find the smallest set of rows which

covers a 1 in each column.

1 1 0 1 1 0

1 1 1 0 0 0

256a 51

0 1 0 1 1 0

0 1 1 1 0 1

1 0 1 1 0 1

Covering set

ma es coverng

set.

A row (column) covers a second row (column) if

it has ones everywhere the second does.

1 0 1 1 0 covers 1 0 0 1 0

1 0

does not

256a 52

0

1

1

0

0

1

cover

If a column a covers column b, column a may be eliminated

A

B

C

D

E

U V W X Y Z

1 0 0 1 1 0

1 1 1 0 0 0

0 1 0 1 1 0

0 1 1 1 0 1

1 0 0 1 0 1

A

B

C

D

E

U W Y Z

1 0 1 0 1 0

1 1 0 0 0 0

0 0 1 0 1 0

0 1 0 1 0 1

1 0 0 1 0 1

256a 53

V covers W

X covers Y

A

B

C

D

U W Y Z

1 0 1 0 1 0

1 1 0 0 0 0

0 0 1 0 1 0

0 1 0 1 0 1

1 0 0 1 0 1

If row a covers row b, row b can be eliminated

A

B

D

E

W Y Z

0 1 0

1 0 0

1 0 1

0 0 1

U

Covers

Y

D covers B and E

256a 54

A covers C

W Y Z

A

D

0 1 0

1 0 1

Smallest covering set : rows A and D

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