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Dimensión M étrica de G rafos. Antonio González Departamento de Matemática Aplicada I Universidad de Sevilla. 28 de noviembre de 2012. Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla. - PowerPoint PPT Presentation
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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
Resolving Sets and Metric Dimension
u3
u2
u1
Resolving Sets and Metric Dimension
u3
u2
u1
Resolving Sets and Metric Dimension
u3
u2
u1
(3,2,1)
Resolving Sets and Metric Dimension
u3
u2
u1
(3,2,1)
Resolving Sets and Metric Dimension
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)
Resolving Sets and Metric Dimension
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
Resolving Sets and Metric Dimension
dim(G) = cardinality of a minimum resolving set
dim(Kn) = n-1 dim(Cn) = 2
dim(Pn) = 1
Problem: Given n coins, each with one of two distinct weights, identify the weight of every coin with the minimum number of weighings.
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}
Problem: Given n coins, each with one of two distinct weights, identify the weight of every coin with the minimum number of weighings.
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
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
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
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
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
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+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+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+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+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+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+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
Affine planesof order k
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
k points in every line
Affine planesof order k
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
k points in every line
Affine planesof order k
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!!!
d(L,Y) ≠ d(L,X)
Finite Geometries
k points in every line
Affine planesof order k
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!!!
d(L,Y) ≠ d(L,X) ≠
Finite Geometries
k points in every line
Affine planesof order k
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!!!
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=
=
Realizabilitydim(G) ≤ dim+(G) ≤ res(G)
a
=
c=
b
[Chartrand et al.,2000]
Realizabilitydim(G) ≤ dim+(G) ≤ res(G)
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:
[Chartrand et al.,2000]
Realizabilitydim(G) ≤ dim+(G) ≤ res(G)=
a
=
b
How many???
[Chartrand et al.,2000]
Realizabilitydim(G) ≤ dim+(G) ≤ res(G)=
a
=
b
How many???
[Chartrand et al.,2000]
Realizabilitydim(G) ≤ dim+(G) ≤ res(G)=
a
=
b
How many???
[Chartrand et al.,2000]
Realizabilitydim(G) ≤ dim+(G) ≤ res(G)=
a
=
b
How many???
[Chartrand et al.,2000]
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).
[Chartrand et al.,2000]
ReconstructionProblem: given c > 0, which are the graphs G such that res(G) = c?
res ≤ 2 [Chartrand,Zhang,2000] Paths and odd cycles.
ReconstructionProblem: given c > 0, which are the graphs G such that res(G) = c?
res ≤ 2
res = 3
[Chartrand,Zhang,2000] Paths and odd cycles.
[Garijo,G.,Márquez,2011] Even cycles plus other 18 graphs.
ReconstructionProblem: given c > 0, which are the graphs G such that res(G) = c?
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!