Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla
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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
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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
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What is a graph?vertices
G=(V,E)edgesn = |V|order
degree4
2
3x
y
d(x,y)=3
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What is a graph?
Complete Graph Kn Cycle Cn
Path Pn Trees
leaves
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Resolving Sets and Metric Dimension
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Resolving Sets and Metric Dimension
u3
u2
u1
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Resolving Sets and Metric Dimension
u3
u2
u1
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Resolving Sets and Metric Dimension
u3
u2
u1
(3,2,1)
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Resolving Sets and Metric Dimension
u3
u2
u1
(3,2,1)
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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)
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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
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Resolving Sets and Metric Dimension
dim(G) = cardinality of a minimum resolving set
dim(Kn) = n-1 dim(Cn) = 2
dim(Pn) = 1
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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}
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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}
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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]
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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))
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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.
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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.
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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.
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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.
![Page 21: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/21.jpg)
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.
![Page 22: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/22.jpg)
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.
![Page 23: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/23.jpg)
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.
![Page 24: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/24.jpg)
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.
![Page 25: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/25.jpg)
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.
![Page 26: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/26.jpg)
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.
![Page 27: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/27.jpg)
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.
![Page 28: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/28.jpg)
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.
![Page 29: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/29.jpg)
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.
![Page 30: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/30.jpg)
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.
![Page 31: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/31.jpg)
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.
![Page 32: Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla](https://reader035.fdocuments.net/reader035/viewer/2022070421/56816254550346895dd29d8e/html5/thumbnails/32.jpg)
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!!!
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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!!!
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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)
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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) ≠
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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))=
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What else?
Partial geometries
Steiner systems
Toroidal grids
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The number of resolving sets of a graph
dim = 2 dim = 5# bases = 1 # bases = 6
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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
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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 ???
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Realizabilitydim(G) ≤ dim+(G) ≤ res(G)
dim(Kn)=n-1
a
=
[Chartrand et al.,2000]
res(Kn)=n-1
c=
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=
Realizabilitydim(G) ≤ dim+(G) ≤ res(G)
a
=
c=
b
[Chartrand et al.,2000]
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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]
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Realizabilitydim(G) ≤ dim+(G) ≤ res(G)=
a
=
b
How many???
[Chartrand et al.,2000]
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Realizabilitydim(G) ≤ dim+(G) ≤ res(G)=
a
=
b
How many???
[Chartrand et al.,2000]
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Realizabilitydim(G) ≤ dim+(G) ≤ res(G)=
a
=
b
How many???
[Chartrand et al.,2000]
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Realizabilitydim(G) ≤ dim+(G) ≤ res(G)=
a
=
b
How many???
[Chartrand et al.,2000]
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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]
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ReconstructionProblem: given c > 0, which are the graphs G such that res(G) = c?
res ≤ 2 [Chartrand,Zhang,2000] Paths and odd cycles.
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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.
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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.
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Thanks!