Find out the difference(s)?

Find out the difference(s)?

Shape 1 Shape 2

Find out the difference(s)?

Graph G1 Graph G2

Connectivity in graphs

Jean Cousty MorphoGraph and Imagery

2011

Outline of the lecture

1 Path

2 Connectivity

3 Algorithms

4 Degrees

Path

Path

Definition

Let G = (E,Γ) be a graph, and let x and y be two vertices in E A path fromx toy (inG) is a sequenceπ = (x0, . . . ,x`)of vertices in E such that

∀i∈ {1, . . . , `}, xi∈Γ(xi−1) x0=x and x`=y

Ifπ= (x₀, . . . ,x<sub>`</sub>) is a path, `is called itslength

2

3

4 1

Exemple

π= (1,2,3)is a path

of length 2

Path

Path

Definition

Let G = (E,Γ) be a graph, and let x and y be two vertices in E A path fromx toy (inG) is a sequenceπ = (x0, . . . ,x`)of vertices in E such that

∀i∈ {1, . . . , `}, xi∈Γ(xi−1) x0=x and x`=y

Ifπ= (x₀, . . . ,x<sub>`</sub>) is a path, `is called itslength

2

3 1

Exemple

π= (1,2,3)is a path

of length 2

Path

Path

Definition

Let G = (E,Γ) be a graph, and let x and y be two vertices in E A path fromx toy (inG) is a sequenceπ = (x0, . . . ,x`)of vertices in E such that

∀i∈ {1, . . . , `}, xi∈Γ(xi−1) x0=x and x`=y

Ifπ = (x₀, . . . ,x<sub>`</sub>) is a path, `is called itslength

2

3

4 1

Exemple

π= (1,2,3)is a path of length 2

Path

Some remarkable paths

A path of length 0 is called a trivial path

A non-trivial pathπ= (x₀, . . . ,x_{`</sub>) is called a circuit ifx<sub>0</sub>=x<sub>`} A pathπ= (x₀, . . . ,x_{`</sub>) is calledelementaryif any two of its vertices are distinct (except possiblyx0 andx<sub>`}): ∀i,j ∈ {0, . . . , `}, i 6=j =⇒ x<sub>i</sub> 6=x<sub>j</sub> (where{i,j} 6={0, `})

2

3

4 1

Exemple

(3)is a trivial path

(1,2,3,1)is a circuit

that is elementary

(1,3,1,2,3,1)is a circuit that is not elementary

Path

Some remarkable paths

A path of length 0 is called a trivial path

A non-trivial path π= (x₀, . . . ,x_{`</sub>) is called a circuit ifx<sub>0</sub>=x<sub>`}

A pathπ= (x₀, . . . ,x_{`</sub>) is calledelementaryif any two of its vertices are distinct (except possiblyx0 andx<sub>`}): ∀i,j ∈ {0, . . . , `}, i 6=j =⇒ x<sub>i</sub> 6=x<sub>j</sub> (where{i,j} 6={0, `})

2

3

4 1

Exemple

(3)is a trivial path (1,2,3,1)is a circuit

that is elementary

(1,3,1,2,3,1)is a circuit that is not elementary

Path

Some remarkable paths

A path of length 0 is called a trivial path

A non-trivial path π= (x₀, . . . ,x_{`</sub>) is called a circuit ifx<sub>0</sub>=x<sub>`} A path π= (x0, . . . ,x`) is calledelementaryif any two of its vertices are distinct (except possibly x0 andx<sub>`</sub>): ∀i,j ∈ {0, . . . , `}, i 6=j =⇒ x<sub>i</sub> 6=x<sub>j</sub> (where{i,j} 6={0, `})

2

3

4 1

Exemple

(3)is a trivial path

(1,2,3,1)is a circuit that is elementary

(1,3,1,2,3,1)is a circuit that is not elementary

Path

Some remarkable paths

A path of length 0 is called a trivial path

A non-trivial path π= (x₀, . . . ,x_{`</sub>) is called a circuit ifx<sub>0</sub>=x<sub>`} A path π= (x0, . . . ,x`) is calledelementaryif any two of its vertices are distinct (except possibly x0 andx<sub>`</sub>): ∀i,j ∈ {0, . . . , `}, i 6=j =⇒ x<sub>i</sub> 6=x<sub>j</sub> (where{i,j} 6={0, `})

2

3

4 1

Exemple

(3)is a trivial path

(1,2,3,1)is a circuit that is elementary

(1,3,1,2,3,1)is a circuit that is not elementary

Path

Basic properties

Property

Any path π from x to y contains an elementary path from x to y

The length of an elementary circuit is less than n (where n=|E|) The length of an elementary path which is not a circuit is less than n−1

Path

Basic properties

Property

Any path π from x to y contains an elementary path from x to y The length of an elementary circuit is less than n (where n =|E|)

The length of an elementary path which is not a circuit is less than n−1

Path

Basic properties

Property

Any path π from x to y contains an elementary path from x to y The length of an elementary circuit is less than n (where n =|E|) The length of an elementary path which is not a circuit is less than n−1

Path

Undirected paths and cycles

Definition

Let G_s = (E,Γ_s) be the symmetric closure of G

Any path in Gs is called anundirected path (in G)

Acycle (in G)is a circuit in Gs that does not pass twice by the same edge

Remark. Ifπ= (x0, . . . ,x`) is an undirected path,

thenπ⁰ = (x_`, . . . ,x₀) is an undirected path (since (E,Γ_s) is a symmetric graph, which implies thatx_i ∈Γ_s(xi−1)⇔x_i−1∈Γ_s(x_i))

Path

Undirected paths and cycles

Definition

Let G_s = (E,Γ_s) be the symmetric closure of G Any path in Gs is called anundirected path (in G)

Acycle (in G)is a circuit in Gs that does not pass twice by the same edge

Remark. Ifπ= (x0, . . . ,x`) is an undirected path,

thenπ⁰ = (x_`, . . . ,x₀) is an undirected path (since (E,Γ_s) is a symmetric graph, which implies thatx_i ∈Γ_s(xi−1)⇔x_i−1∈Γ_s(x_i))

Path

Undirected paths and cycles

Definition

Let G_s = (E,Γ_s) be the symmetric closure of G Any path in Gs is called anundirected path (in G)

A cycle (in G)is a circuit in Gs that does not pass twice by the same edge

Remark. Ifπ= (x0, . . . ,x`) is an undirected path,

thenπ⁰ = (x_`, . . . ,x₀) is an undirected path (since (E,Γ_s) is a symmetric graph, which implies thatx_i ∈Γ_s(xi−1)⇔x_i−1∈Γ_s(x_i))

Path

Undirected paths and cycles

Definition

Let G_s = (E,Γ_s) be the symmetric closure of G Any path in Gs is called anundirected path (in G)

A cycle (in G)is a circuit in Gs that does not pass twice by the same edge

Remark. Ifπ= (x0, . . . ,x`) is an undirected path,

thenπ⁰ = (x_`, . . . ,x₀) is an undirected path (since (E,Γ_s) is a symmetric graph, which implies thatx_i ∈Γ_s(xi−1)⇔x_i−1 ∈Γ_s(x_i))

Path

Illustration: path and undirected path, circuit and cycles

2

3

4 1

Exemple

(1,2,1)is not a path (1,2,3,1,2,4)is a path (4,2,1)is an undirected path (1,3,2,1)is not a circuit (1,3,1)is not a cycle

(1,2,3)is a path (4,2,1)is not a path (1,2,3,1)is a circuit (1,3,2,1)is a cycle

Connectivity

Connected component

Definition

Let x ∈E . The connected component of G containingx is the subset C_x of E defined by:

C_x ={y ∈E |there exists an undirected path from x to y}

Exemple

0 1

0

1 01

0 1 0 1

0 1 0000

1111 0 01 01

1 00

11

00 11

00 11 0 1

00 11 00 11

1 2

3

4

6 7 8

5 C₁ ={1,2,3,4}=C₂ =

C₃ =C₄ C₅ ={5}

C₆ =C₇=C₈ ={6,7,8}

Connectivity

Connected component

Definition

Let x ∈E . The connected component of G containingx is the subset C_x of E defined by:

C_x ={y ∈E |there exists an undirected path from x to y}

Exemple

0 1

0 1

0 1 0 1 0 1

0 1 0000

1111 0 01 01

1 00

11

00 11

00 11 0 1

00 11 00 11

1 2

3

4

6 7 8

5 C₁ ={1,2,3,4}=C₂ =

C₃ =C₄ C₅ ={5}

C₆ =C₇ =C₈={6,7,8}

Connectivity

Connected component as equivalence classes

Property

1 ∀x∈E , x ∈C_x (reflexivity)

2 ∀x,y ∈E , y ∈Cx =⇒ x ∈Cy (symmetry)

3 ∀x,y,z ∈E , [y ∈Cx and z ∈Cy] =⇒ z ∈Cx (transitivity)

Proof.

1 ∀x∈E, (x) is a (trivial) undirected path, thusx ∈C_x

2 y ∈C_x =⇒ ∃an undirected pathπ = (x₀, . . . , ,x_`) fromx toy

=⇒ π⁰ = (x`, . . . ,x0) is an undirected path fromy tox

=⇒ x ∈Cy

3 [y ∈Cx andz ∈Cy] =⇒ [∃ an undirected pathπ = (x0, . . . ,x<sub>`</sub>) from x to y and∃an undirected path π<sup>0</sup>= (y<sub>0</sub>, . . . ,y<sub>m</sub>) fromy to z] =⇒ π<sup>00</sup>= (x0, . . . ,x`,y1, . . . ,ym) is an undirected path

Connectivity

Strongly connected component

Definition

Let x ∈E . The strongly connected component (ofG) containing x is the subset C_x⁰ of E defined by

C_x⁰ ={y ∈E | ∃a path from x to y and ∃a path from y to x}

Exemple

0 1

0

1 01

0 1 0 1

0 1 0000

1111 0 01 01

1 00

11

00 11

00 11 0 1

00 11 00 11

1 2

3

4

6 7 8

5 C₁⁰ ={1,2,3}=C₂⁰ =C₃⁰

C₄⁰ ={4} ; C₅⁰ ={5} C₆⁰ ={6} ; C₇⁰ ={7} C₈⁰ ={8}

Connectivity

Strongly connected component

Definition

Let x ∈E . The strongly connected component (ofG) containing x is the subset C_x⁰ of E defined by

C_x⁰ ={y ∈E | ∃a path from x to y and ∃a path from y to x}

Exemple

0 1

0 1 0 1

0 1 0000

1111 0 01 01

1 00

11

0 1

00 11 00 11

1 2

3

7 8

5 C₁⁰ ={1,2,3}=C₂⁰ =C₃⁰

C₄⁰ ={4} ; C₅⁰ ={5}

C₆⁰ ={6} ; C₇⁰ ={7}

C⁰ ={8}

Connectivity

Strongly connected components as equivalence classes?

Exercise. Are the strongly connected components of a graphs equivalence classes? Proof or counter-example?

Algorithms

Connected and strongly connected components computation

Method

We will first study a characterization of connected components and of strongly connected components based on morphological dilation This will allow us to propose efficient algorithm to compute them

Algorithms

Iterated operators

Definition

Let γ be an operator and i ∈N

We denote by γⁱ the operator defined by

1 γⁱ=γγⁱ⁻¹

2 γ⁰=Id (i.e. ∀X ⊆E, γ⁰(X) =X )

Exemple (iterated dilation)

1 δ⁰_Γ(X) =X

2 δ¹_Γ(X) =δ_Γ(δ⁰_Γ(X)) =δ_Γ(X)

3 δ²_Γ(X) =δ_Γ(δ¹_Γ(X)) =δ_Γ(δ_Γ(X))

4 δ³_Γ(X) =δ_Γ(δ²_Γ(X)) =δ_Γ(δ_Γ(δ_Γ(X)))

5 . . .

6 δⁱ_Γ(X) =δ_Γ(δⁱ_Γ⁻¹(X)) =δ_Γ(. . . δ_Γ(X). . .)

Algorithms

Iterated operators

Definition

Let γ be an operator and i ∈N

We denote by γⁱ the operator defined by

1 γⁱ=γγⁱ⁻¹

2 γ⁰=Id (i.e. ∀X ⊆E, γ⁰(X) =X )

Exemple (iterated dilation)

1 δ⁰_Γ(X) =X

2 δ¹_Γ(X) =δ_Γ(δ⁰_Γ(X)) =δ_Γ(X)

3 δ²_Γ(X) =δ_Γ(δ¹_Γ(X)) =δ_Γ(δ_Γ(X))

4 δ³_Γ(X) =δ_Γ(δ²_Γ(X)) =δ_Γ(δ_Γ(δ_Γ(X)))

5 . . .

Algorithms

Illustration: iterated dilation

1 2

3

4

6 7 8

5

Exemple

X =δ⁰_Γ(X) ={1}

δ¹_Γ(X) ={2} δ²_Γ(X) ={3,4}

Algorithms

Illustration: iterated dilation

1 2

3

4

6 7 8

5

Exemple

X =δ⁰_Γ(X) ={1}

δ¹_Γ(X) ={2}

δ²_Γ(X) ={3,4}

Algorithms

Illustration: iterated dilation

1 2

3

4

6 7 8

5

Exemple

X =δ⁰_Γ(X) ={1}

δ¹_Γ(X) ={2}

δ²_Γ(X) ={3,4}

Algorithms

Iterated dilation and paths of given length

Property

Let x ∈E and i ∈N

The two following equalities hold true

δⁱ_Γ({x}) ={y ∈E | ∃a path from x to y of length i}

δⁱ_Γ−1({x}) ={y∈E | ∃a path from y to x of length i}

Definition

We define for any X ⊆E and any p∈N δˆ^p_Γ(X) =∪{δ_Γⁱ(X)|i∈ {0, . . . ,p}}

The setδˆ_Γ^p(X) is called the pre-closure ofX of rankp

Algorithms

Iterated dilation and paths of given length

Property

Let x ∈E and i ∈N

The two following equalities hold true

δⁱ_Γ({x}) ={y ∈E | ∃a path from x to y of length i}

δⁱ_Γ−1({x}) ={y∈E | ∃a path from y to x of length i}

Definition

We define for any X ⊆E and any p∈N δˆ^p_Γ(X) =∪{δⁱ_Γ(X)|i∈ {0, . . . ,p}}

The setδˆ_Γ^p(X) is called the pre-closure ofX of rankp

Algorithms

Iterated dilation and paths of given length

Property

Let x ∈E and i ∈N

The two following equalities hold true

δⁱ_Γ({x}) ={y ∈E | ∃a path from x to y of length i}

δⁱ_Γ−1({x}) ={y∈E | ∃a path from y to x of length i}

Definition

We define for any X ⊆E and any p∈N δˆ^p_Γ(X) =∪{δⁱ_Γ(X)|i∈ {0, . . . ,p}}

The set δˆ_Γ^p(X) is called the pre-closure ofX of rankp

Algorithms

Transitive closure and paths

Corollary

Let x ∈E

The two following equalities hold true

δˆ^p_Γ({x}) ={y∈E | ∃a path from x to y of length ≤p}

δ^p_Γˆ−1({x}) ={y∈E | ∃a path from y to x of length ≤p}

Definition

Let x ∈E , the (transitive) closure of {x} is the set δˆ^∞_Γ ({x}) ={y ∈E | ∃a path from x to y}

Exercise. Prove that ˆδ_Γ^∞({x}) =δ_Γⁿ⁻¹ˆ ({x}) (wheren=|E|)

Algorithms

Transitive closure and paths

Corollary

Let x ∈E

The two following equalities hold true

δˆ^p_Γ({x}) ={y∈E | ∃a path from x to y of length ≤p}

δ^p_Γˆ−1({x}) ={y∈E | ∃a path from y to x of length ≤p}

Definition

Let x ∈E , the (transitive) closure of {x} is the set δˆ^∞_Γ ({x}) ={y ∈E | ∃a path from x to y}

Exercise. Prove that ˆδ_Γ^∞({x}) =δ_Γⁿ⁻¹ˆ ({x}) (wheren=|E|)

Algorithms

Transitive closure and paths

Corollary

Let x ∈E

The two following equalities hold true

δˆ^p_Γ({x}) ={y∈E | ∃a path from x to y of length ≤p}

δ^p_Γˆ−1({x}) ={y∈E | ∃a path from y to x of length ≤p}

Definition

Let x ∈E , the (transitive) closure of {x} is the set δˆ^∞_Γ ({x}) ={y ∈E | ∃a path from x to y}

Exercise. Prove that ˆδ_Γ^∞({x}) =δ_Γⁿ⁻¹ˆ ({x}) (where n=|E|)

Algorithms

Illustration: transitive closure

1 2

3

4

6 7 8

5

Exemple

X =δ⁰_Γ(X) ={1}

δ¹_Γ(X) ={2} δ²_Γ(X) ={3,4}

X = ˆδ⁰_Γ(X) ={1}

δˆ_Γ¹(X) ={1,2}

δˆ_Γ²(X) ={1,2,3,4}= ˆδ_Γ⁷(X)

Algorithms

Illustration: transitive closure

1 2

3

4

6 7 8

5

Exemple

X =δ⁰_Γ(X) ={1}

δ¹_Γ(X) ={2}

δ²_Γ(X) ={3,4}

X = ˆδ⁰_Γ(X) ={1}

δˆ_Γ¹(X) ={1,2}

δˆ_Γ²(X) ={1,2,3,4}= ˆδ_Γ⁷(X)

Algorithms

Illustration: transitive closure

1 2

3

4

6 7 8

5

Exemple

X =δ⁰_Γ(X) ={1}

δ¹_Γ(X) ={2}

δ²(X) ={3,4}

X = ˆδ⁰_Γ(X) ={1}

δˆ_Γ¹(X) ={1,2}

δˆ²(X) ={1,2,3,4}= ˆδ⁷(X)

Algorithms

Transitive closure and connected components

Property

Let x ∈E . Let Cx and C_x⁰ be respectively the connected

components and the strongly connected components containing x

The two following equalities hold true C_x⁰ =δ_Γⁿ⁻¹ˆ ({x})∩δ_Γⁿ⁻¹ˆ−1({x}) Cx =δ_Γⁿ⁻¹ˆ

s ({x}) =δⁿ⁻¹_Γˆ

s ({x})

where n=|E|andΓs is the symmetric closure ofΓ

Important remark

To compute the connected and strongly connected components containingx, it is thus sufficient to compute the transitive closure of{x} for the graphs (E,Γ), (E,Γ⁻¹) and (E,Γ_s)

Algorithms

Transitive closure and connected components

Property

Let x ∈E . Let Cx and C_x⁰ be respectively the connected

components and the strongly connected components containing x The two following equalities hold true

C_x⁰ =δ_Γⁿ⁻¹ˆ ({x})∩δ_Γⁿ⁻¹ˆ−1({x}) Cx =δ_Γⁿ⁻¹ˆ

s ({x}) =δⁿ⁻¹_Γˆ

s ({x})

where n=|E|andΓs is the symmetric closure ofΓ

Important remark

To compute the connected and strongly connected components containingx, it is thus sufficient to compute the transitive closure of{x} for the graphs (E,Γ), (E,Γ⁻¹) and (E,Γ_s)

Algorithms

Transitive closure and connected components

Property

Let x ∈E . Let Cx and C_x⁰ be respectively the connected

components and the strongly connected components containing x The two following equalities hold true

C_x⁰ =δ_Γⁿ⁻¹ˆ ({x})∩δ_Γⁿ⁻¹ˆ−1({x}) Cx =δ_Γⁿ⁻¹ˆ

s ({x}) =δⁿ⁻¹_Γˆ

s ({x})

where n=|E|andΓs is the symmetric closure ofΓ

Important remark

To compute the connected and strongly connected components containing x, it is thus sufficient to compute the transitive closure of {x} for the graphs (E,Γ), (E,Γ⁻¹) and (E,Γ_s)

Algorithms

Naive algorithm for the transitive closure of {x }

Algorithm TRANS NAIV ( Data: (E,Γ),x ∈E ;

Results: Z =δ_Γⁿ⁻¹ˆ ({x}))

X :={x} ;Y :=∅ ;Z :={x} ; For each i from 1 to n−1 do

Y := DIL((E,Γ),X) ; /*Y =δ_Γⁱ({x}) */

Z :=Z∪Y ; /*Z = ˆδ_Γⁱ({x}) */

X :=Y;Y :=∅;

Complexity

By using the algorithm DIL studied during practical session # 1 The complexity of TRANS NAIV is

O(n²+nm)(where n=|E|and m=|Γ|)

Algorithms

Naive algorithm for the transitive closure of {x }

Algorithm TRANS NAIV ( Data: (E,Γ),x ∈E ;

Results: Z =δ_Γⁿ⁻¹ˆ ({x}))

X :={x} ;Y :=∅ ;Z :={x} ; For each i from 1 to n−1 do

Y := DIL((E,Γ),X) ; /*Y =δ_Γⁱ({x}) */

Z :=Z∪Y ; /*Z = ˆδ_Γⁱ({x}) */

X :=Y;Y :=∅;

Complexity

By using the algorithm DIL studied during practical session # 1 The complexity of TRANS NAIV is

O(n²+nm)(where n=|E|and m=|Γ|)

Algorithms

Linear-time algorithm for the transitive closure of {x }

Algorithm TRANS ( Data: (E,Γ),x∈E ;

Result: Z =δ_Γⁿ⁻¹ˆ ({x})) X :={x} ;Y :=∅ ;Z :={x} ;

For eachi from 1 to n−1do While ∃y ∈X do

X :=X \ {y} ; For eachz ∈Γ(y)do

Ifz∈/Z thenY :=Y ∪ {z};Z :=Z ∪ {z};

X :=Y;Y :=∅;

Complexity

If X and Y are represented by LLs and if Z is represented by a BA The time-complexity of TRANS is linear

O(n+m)(where n=|E|and m=|Γ|)

Algorithms

Linear-time algorithm for the transitive closure of {x }

Algorithm TRANS ( Data: (E,Γ),x∈E ;

Result: Z =δ_Γⁿ⁻¹ˆ ({x})) X :={x} ;Y :=∅ ;Z :={x} ;

For eachi from 1 to n−1do While ∃y ∈X do

X :=X \ {y} ; For eachz ∈Γ(y)do

Ifz∈/Z thenY :=Y ∪ {z};Z :=Z ∪ {z};

X :=Y;Y :=∅;

Complexity

If X and Y are represented by LLs and if Z is represented by a BA The time-complexity of TRANS is linear

O(n+m)(where n=|E|and m=|Γ|)

Algorithms

Linear-time algorithm for the transitive closure of {x }

Algorithm TRANS can be further simplified without changing its complexity:

Algorithm TRANS ( Data: (E,Γ),x∈E ;

Result: Z =δ_Γⁿ⁻¹ˆ ({x}))

X :={x} ;Z :={x};

While ∃y ∈X do X :=X \ {y};

For eachz ∈Γ(y)do

Ifz∈/Z then X :=X ∪ {z};Z :=Z∪ {z};

Algorithms

Computing the strongly connected component containing x

Algorithm SCC ( Data: (E,Γ),x ∈E ;

Result: Z =C_x⁰)

X := TRANS((E,Γ),x);

Γ⁻¹ := SYM 1(E,Γ);

Y := TRANS((E,Γ⁻¹), x) Z :=X ∩Y;

Complexity

Using

the linear-time algorithm SYM 1 (first lecture) the linear-time TRANS

The time-complexity of SCC is linear: O(n+m)(where n=|E|and m=|Γ|)

Algorithms

Computing the strongly connected component containing x

Algorithm SCC ( Data: (E,Γ),x ∈E ;

Result: Z =C_x⁰)

X := TRANS((E,Γ),x);

Γ⁻¹ := SYM 1(E,Γ);

Y := TRANS((E,Γ⁻¹), x) Z :=X ∩Y;

Complexity

Using

the linear-time algorithm SYM 1 (first lecture) the linear-time TRANS

The time-complexity of SCC is linear:

Algorithms

Computing the connected component containing x

Algorithm CC ( Data: (E,Γ),x ∈E ;

Result: Y =Cx)

Γ⁻¹ := SYM 1(E,Γ);

Γ_s := Γ∪Γ⁻¹ ;

Y := TRANS((E,Γ_s),x)

Complexity

Using

the linear-time algorithm SYM 1 (first lecture) the linear-time TRANS

The time-complexity of CC is linear: O(n+m)(where n=|E|and m=|Γ|)

Algorithms

Computing the connected component containing x

Algorithm CC ( Data: (E,Γ),x ∈E ;

Result: Y =Cx)

Γ⁻¹ := SYM 1(E,Γ);

Γ_s := Γ∪Γ⁻¹ ;

Y := TRANS((E,Γ_s),x)

Complexity

Using

the linear-time algorithm SYM 1 (first lecture) the linear-time TRANS

The time-complexity of CC is linear:

O(n+m)(where n=|E|and m=|Γ|)

Degrees

Application of graphs properties

Problem

In a charity diner, is there always two people having the same number of friends who are present in the diner?

Can we draw in the plan five distinct lines such that any of them has exactly three intersection points with the others?

In order to answer these questions read first the two next slides!

Degrees

Application of graphs properties

Problem

In a charity diner, is there always two people having the same number of friends who are present in the diner?

Can we draw in the plan five distinct lines such that any of them has exactly three intersection points with the others?

In order to answer these questions read first the two next slides!

Degrees

Degree

Let G = (E,Γ) be a graph and letx ∈E

The outer degree of x (for G )is the valued⁺(x) =|Γ(x)|

The inner degree of x (for G )is the valued⁻(x) =|Γ⁻¹(x)|

The degree of x (for G )is the valued(x) =d⁺(x) +d⁻(x) Let G⁰ = (E,Γ) be an undirected graph

The degree of x for G⁰ is the number d(x) of edges that are adjacent to x

Degrees

Degree: exercise

Prove that the two following propositions hold true The sum of the degrees of the vertices of a graph is even

In any graph, there is an even number of vertices whose degree is odd