Lecture 1

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Lecture 1

LECTURE 6

Graphs seen as convexity spaces

Graph-theoretical and classical parameters

Ignacio M Pelayo

UNIVERSITAT POLITÈCNICA DE CATALUNYA

BARCELONA, SPAIN


Graph Convexity Spaces

Graph-theoretical Parameters (def., ex.)

Independency Parameters (def., ex.)

Independency parameters (prop.)


C g

C c

C tf

C t

C m


[S] c

[S] g

[S] m

[S] t

[S] tf


10

7 8 9

5 6

S={2,8}

1

2 3 4

[S] g ={1,2,6,8} [S] m = [S] g U {3,4}

[S] t = [S] m U {9} [S] tf = [S] m U {5,7}

[S] c = V-10


Graph Convexity Spaces

Graph-theoretical Parameters (def.,ex.)

Independency Parameters (def., ex.)

Independency parameters (prop.)


PETERSEN GRAPH

n

m


D

g

10

15

3

2

5

con

gn

hn

gin

mn

5

4

3

3

3

a

1

e

5

b

S={a,c,4,5} is geodetic

2

4

3

c


PETERSEN GRAPH

S={a,c,5}

I[S]={a,b,c,3,5,e}

con

gn

hn

gin

mn

I 2 [S]=V-4

5

4

3

3

3

I 3 [S]=V

a

gin(S)=3

e

5

1

b

S={a,c,5} is a hull set

2

3

S={a,c,5} is monophonic (easy)

d

4

c


Graph Convexity Spaces

Graph-theoretical Parameters (def., ex.)

Independency Parameters (def., ex.)

Independency parameters (prop.)


[1,3]={1,4,3} [1,2]={1,2} [2,3]={2,5,3}

4

3

3

5

1

1 2

2

1,2,3]={1,2,3,4,5} [1,2,3] = [1,3]

3

[1,2,3] = [1,2]

NO

NO

convex indep.

5

[1,2,3] = [2,3]

NO

[1,3] U [1,2] U [2,3] = [1,2,3]

[1,3] I [1,2] I [2,3] = 0

SI

SI

redundant

Helly indep.


Graph Convexity Spaces

Graph-theoretical Parameters (def., ex.)

Independency Parameters (def., ex.)

Independency parameters (prop.)

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