seminario phd del instituto de matemáticas d e la universidad de sevilla

Seminario PHD del Instituto de Matemáticas de la Universidad de Sevilla

Dimensión Métrica de Grafos

Antonio González

Departamento de Matemática Aplicada I

Universidad de Sevilla

28 de noviembre de 2012

Problem: Given n coins, each with one of two distinct weights, identify the weight of every coin with the minimum number of weighings.

{1,2,3} {2,3,4} {3,4}

1 false coin 2 false coins 2 false coins

1 2

3 4

SOLUTION: METRIC DIMENSION OF THE HYPERCUBE!!!

n = 4

What is a graph?vertices

G=(V,E)edgesn = |V|order

degree4

2

3x

y

d(x,y)=3

What is a graph?

Complete Graph Kn Cycle Cn

Path Pn Trees

leaves

Resolving Sets and Metric Dimension


u3

u2

u1


u3

u2

u1

(3,2,1)


u3

u2

u1

(3,2,1)

(0,3,3)

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

(3,0,3)

(3,3,0)

(2,3,1)

(3,1,2)

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


u2

u1 (0,3)

(2,1)(1,2)

(3,0)

(3,3)

(2,3)

(3,1)

(2,2)(1,3)

(3,2)

dim(G) = cardinality of a minimum resolving set

METRIC

BASIS


dim(G) = cardinality of a minimum resolving set

dim(Kn) = n-1 dim(Cn) = 2

dim(Pn) = 1


n = 4

This is the hypercube Qn !!!V(Qn) = Subsets of {1,2,3,…,n} E(Qn) = { {U,V} : |U ∆ V|= 1 } d( U , V ) = | U ∆ V| = |U| + |V| - 2|U ∩ V|

X

How to determine a possible situation X ?

X

{1} {2} {3} {4}

{1,3} {1,4} {2,3} {2,4}{1,2} {3,4}

{1,2,3} {1,2,4} {1,3,4} {2,3,4}

{1,2,3,4}

Ø

{1,3}

{1,2}{2,3,4}


n = 4

This is the hypercube Qn !!!V(Qn) = Subsets of {1,2,3,…,n} E(Qn) = { {U,V} : |U ∆ V|= 1 } d( U , V ) = | U ∆ V| = |U| + |V| - 2|U ∩ V|

How to determine a possible situation X ?

dim(Qn) + 1

{1} {2} {3} {4}

{1,3} {1,4} {2,3} {2,4}{1,2} {3,4}

{1,2,3} {1,2,4} {1,3,4} {2,3,4}

{1,2,3,4}

Ø

X

{1,3}

X

S can determine X !!!

d(X,Si) for every Si є S

d(X,Si) = |X| + |Si| - 2|X ∩ Si|

S resolving set of Qn

?{1,2,3,4}

Problem: Given n coins, each with one of two distinct weights, identify the weight of every coin with the minimum number of weighings. dim(Qn) + 1

[Erdős,Rényi,1963] [Lindström,1964]

Problem: Given n coins, each with one of two distinct weights, identify the weight of every coin with the minimum number of k-weighings if we have exactly k true coins.n = 5k = 2

This is the Johnson graph J(n,k) !!!

V(J(n,k)) = k-subsets of {1,2,3,…,n} E(Qn) = { {U,V} : |U ∆ V|= 2 }

d( U , V ) = ½| U ∆ V| = ½(|U| + |V| - 2|U ∩ V|)= k - |U ∩ V|

{1,4}

{2,3}

{2,4}

{1,2}

{3,4}{1,5}

{3,5}{4,5}

{2,5} {1,3}

dim(J(n,k))

n = 5k = 2

This is the Johnson graph J(n,k) !!!

V(J(n,k)) = k-subsets of {1,2,3,…,n} E(Qn) = { {U,V} : |U ∆ V|= 2 }

d( U , V ) = ½| U ∆ V| = ½(|U| + |V| - 2|U ∩ V|)= k - |U ∩ V|

{1,4}

{2,3}

{2,4}

{1,2}

{3,4}{1,5}

{3,5}{4,5}

{2,5} {1,3}

dim(J(n,k))

Problem: Given n coins, each with one of two distinct weights, identify the weight of every coin with the minimum number of k-weighings if we have exactly k true coins.

dim(J(n,k))

J(6,2) J(7,2) J(8,2)

J(6,3) J(7,3) J(8,3)

Can we find any tool to approach the metric dimension of these graphs?

FINITE GEOMETRIES

Problem: Given n coins, each with one of two distinct weights, identify the weight of every coin with the minimum number of k-weighings if we have exactly k true coins.

Finite Geometries

k+1 points in every line

Projective planesof order k

A finite geometry (P,L) is a finite set P called points together with a non-empty collection L of subsets of P called lines.

Finite Geometries

k+1 points in every line k points in every line

Projective planesof order k

Affine planesof order k

A finite geometry (P,L) is a finite set P called points together with a non-empty collection L of subsets of P called lines.

Finite Geometries

k points in every line


X Y

3 4

8

56

9

1

2

7

J(9,3)

Given two vertices X,Y є V(J(n,k)), is there any line L є L distinguishing them?

There exist k+1 distinct lines through every point!!!

Finite Geometries



X Y

3 4

8

56

9

1

2

7

J(9,3)



d(L,Y) ≠ d(L,X)

Finite Geometries



X Y

3 4

8

56

9

1

2

7

J(9,3)



d(L,Y) ≠ d(L,X) ≠

Finite Geometries



X Y

3 4

8

56

9

1

2

7

J(9,3)



d(L,Y) ≠ d(L,X) ≠

[Cáceres,Garijo,G.,Márquez,Puertas,2011]

Proposition: If k ≥ 3 is a prime power, then

dim(J(k2,k)) ≤ k2 + k and dim(J(k2+k+1,k+1)) ≤ k2 + k+1.

Proposition: If n ≥ 3, then

dim(J(n,2))=

What else?

Partial geometries

Steiner systems

Toroidal grids

The number of resolving sets of a graph

dim = 2 dim = 5# bases = 1 # bases = 6

Graphs with “many” metric bases

dim ≤ 2

Open Problem [Chartrand,Zhang,2000]: Characterize the graphs G such that every subset of size dim(G) is a basis.

[Chartrand,Zhang,2000] Complete graphs and odd cycles.

dim > 2 [Garijo,G.,Márquez,2011] Complete graphs.

K1K2

C3C5

Upper Dimension and Resolving Number

(1,1,2,2,3)

(1,1,2,2,3)

dim+(G) = 4 res(G) = 6

UPPER

BASISdim(G) ≤ dim+(G) ≤ res(G)

res(G)= minimum k such that every k-subset is a resolving set.

dim+(G)= maximum size of a minimal resolving set

Realizability ???

Realizabilitydim(G) ≤ dim+(G) ≤ res(G)

dim(Kn)=n-1

a

=

[Chartrand et al.,2000]

res(Kn)=n-1

c=

=


a

=

c=

b



Conjecture: For every pair a,b of integers with 2≤a≤b, there exists a conected graph G such that dim(G)=a and dim+(G)=b.

=

a

=

b

It is true!!! [Garijo,G.,Márquez,2011]

Theorem:


Realizabilitydim(G) ≤ dim+(G) ≤ res(G)=

a

=

b

How many???


Realizabilitydim(G) ≤ dim+(G) ≤ res(G)=

a

=

b

How many???

c=

Theorem:[Garijo,G.,Márquez] Given c>3, the set of graphs with resolving number c is finite.

QUESTION (2): RECONSTRUCTION!!!

QUESTION (1): Realization of triples (a,b,c).


ReconstructionProblem: given c > 0, which are the graphs G such that res(G) = c?

res ≤ 2 [Chartrand,Zhang,2000] Paths and odd cycles.


res ≤ 2

res = 3

[Chartrand,Zhang,2000] Paths and odd cycles.

[Garijo,G.,Márquez,2011] Even cycles plus other 18 graphs.


res ≤ 2

res = 3

Open problem: Reconstruction of trees.

[Chartrand,Zhang,2000] Paths and odd cycles.

[Garijo,G.,Márquez,2011] Even cycles plus other 18 graphs.

Thanks!

seminario phd del instituto de matemáticas d e la universidad de sevilla

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