Seminario PHD del Instituto de Matemáticas d e la Universidad de Sevilla

1. DimensiónMétrica de Grafos • Antonio González • Departamento de MatemáticaAplicada I • Universidad de Sevilla 28 de noviembre de 2012 Seminario PHD del Instituto de Matemáticas de la Universidad de Sevilla

2. Problem:Givenncoins, eachwithone of twodistinctweights, identifytheweight of every coin with theminimumnumber of weighings. n = 4 2 1 {2,3,4} {1,2,3} {3,4} 2 false coins 1 false coin 2 false coins 4 3 SOLUTION: METRIC DIMENSION OF THE HYPERCUBE!!!

3. Whatis a graph? G=(V,E) vertices edges order n = |V| y x 3 2 d(x,y)=3 4 degree

4. Whatis a graph? CycleCn Complete GraphKn PathPn Trees leaves

5. Resolving Sets and MetricDimension

6. Resolving Sets and MetricDimension u3 u2 u1

7. Resolving Sets and MetricDimension u3 u2 u1

8. Resolving Sets and MetricDimension (3,2,1) u3 u2 u1

9. Resolving Sets and MetricDimension (3,2,1) u3 u2 u1

10. Resolving Sets and MetricDimension (3,1,2) (3,3,0) (3,2,1) u3 (3,0,3) (2,3,1) u2 (2,2,2) (1,3,2) (2,1,3) (1,2,3) u1 (0,3,3)

11. Resolving Sets and MetricDimension dim(G) = cardinality of a minimumresolving set (3,1) (3,3) (3,2) (3,0) (2,3) u2 (2,2) (1,3) (2,1) METRIC BASIS (1,2) u1 (0,3)

12. Resolving Sets and MetricDimension dim(G) = cardinality of a minimumresolving set dim(Cn) = 2 dim(Kn) = n-1 dim(Pn) = 1

13. Problem:Givenncoins, eachwithone of twodistinctweights, identifytheweight of every coin with theminimumnumber of weighings. Ø {1} {2} {3} {4} n = 4 Howto determine a possiblesituation X ? {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} {1,3} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} X X {2,3,4} {1,2,3,4} ThisisthehypercubeQn!!! 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|

14. Problem:Givenncoins, eachwithone of twodistinctweights, identifytheweight of every coin with theminimumnumber of weighings. dim(Qn) + 1 n = 4 Howto determine a possiblesituation X ? Ø S resolving set of Qn {1} {2} {3} {4} S can determine X !!! {1,2} {1,3} {1,4} {1,3} {2,3} {2,4} {3,4} ? d(X,Si) forevery Si єS {1,2,3} {1,2,4} {1,3,4} {2,3,4} X X {1,2,3,4} {1,2,3,4} d(X,Si) = |X| + |Si| - 2|X ∩ Si| ThisisthehypercubeQn!!! 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|

15. Problem:Givenncoins, eachwithone of twodistinctweights, identifytheweight of every coin with theminimumnumber of weighings. dim(Qn) + 1 [Erdős,Rényi,1963] [Lindström,1964]

16. Problem:Givenncoins, eachwithone of twodistinctweights, identifytheweight of every coin with theminimumnumber of k-weighingsifwehaveexactlyk true coins. n = 5 k = 2 {1,2} dim(J(n,k)) {3,5} {3,4} {4,5} {1,5} {2,3} Thisisthe Johnson graph J(n,k) !!! {1,4} {2,4} {1,3} {2,5} 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|

17. Problem:Givenncoins, eachwithone of twodistinctweights, identifytheweight of every coin with theminimumnumber of k-weighingsifwehaveexactlyk true coins. n = 5 k = 2 {1,2} dim(J(n,k)) {3,5} {3,4} {4,5} {1,5} {2,3} Thisisthe Johnson graph J(n,k) !!! {1,4} {2,4} {1,3} {2,5} 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|

18. Problem:Givenncoins, eachwithone of twodistinctweights, identifytheweight of every coin with theminimumnumber of k-weighingsifwehaveexactlyk true coins. J(7,2) J(8,2) J(6,2) Can wefindanytooltoapproachthemetricdimension of thesegraphs? dim(J(n,k)) J(6,3) J(7,3) J(8,3) FINITE GEOMETRIES

19. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk k+1 points in every line

20. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk k+1 points in every line

21. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk k+1 points in every line

22. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk k+1 points in every line

23. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk k+1 points in every line

24. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk k+1 points in every line

25. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk Affine planes of orderk k+1 points in every line k points in every line

26. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk Affine planes of orderk k+1 points in every line k points in every line

27. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk Affine planes of orderk k+1 points in every line k points in every line

28. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk Affine planes of orderk k+1 points in every line k points in every line

29. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk Affine planes of orderk k+1 points in every line k points in every line

30. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk Affine planes of orderk k+1 points in every line k points in every line

31. FiniteGeometries A finitegeometry(P,L) is a finite set Pcalledpointstogetherwith a non-emptycollectionL of subsets of Pcalledlines. Projective planes of orderk Affine planes of orderk k+1 points in every line k points in every line

32. FiniteGeometries GiventwoverticesX,Yє V(J(n,k)), isthereany line Lє Ldistinguishingthem? Affine planes of orderk J(9,3) 1 4 3 2 5 Y X 6 Thereexistk+1distinctlinesthrougheverypoint!!! 8 9 7 k points in every line

33. FiniteGeometries GiventwoverticesX,Yє V(J(n,k)), isthereany line Lє Ldistinguishingthem? Affine planes of orderk J(9,3) 1 4 3 2 5 Y X 6 Thereexistk+1distinctlinesthrougheverypoint!!! 8 9 7 k points in every line

34. FiniteGeometries GiventwoverticesX,Yє V(J(n,k)), isthereany line Lє Ldistinguishingthem? Affine planes of orderk J(9,3) 1 d(L,Y) ≠ d(L,X) 4 3 2 5 Y X 6 Thereexistk+1distinctlinesthrougheverypoint!!! 8 9 7 k points in every line

35. FiniteGeometries GiventwoverticesX,Yє V(J(n,k)), isthereany line Lє Ldistinguishingthem? Affine planes of orderk J(9,3) 1 ≠ d(L,Y) ≠ d(L,X) 4 3 2 5 Y X 6 Thereexistk+1distinctlinesthrougheverypoint!!! 8 9 7 k points in every line

36. FiniteGeometries GiventwoverticesX,Yє V(J(n,k)), isthereany line Lє Ldistinguishingthem? [Cáceres,Garijo,G.,Márquez,Puertas,2011] Proposition: Ifk ≥ 3 is a prime power, then dim(J(k2,k)) ≤ k2 + k and dim(J(k2+k+1,k+1)) ≤ k2 + k+1. Affine planes of orderk J(9,3) Proposition: Ifn≥ 3, then dim(J(n,2))= 1 ≠ d(L,Y) ≠d(L,X) 4 3 2 5 Y X 6 Thereexistk+1distinctlinesthrougheverypoint!!! 8 9 7 k points in every line

37. Whatelse? Steiner systems Toroidalgrids Partialgeometries

38. Thenumber of resolving sets of a graph dim = 2 dim = 5 # bases = 1 # bases = 6

39. Graphswith “many” metric bases Open Problem [Chartrand,Zhang,2000]: CharacterizethegraphsGsuchthateverysubset of sizedim(G) is a basis. dim ≤ 2 [Chartrand,Zhang,2000] Complete graphs and oddcycles. dim > 2 [Garijo,G.,Márquez,2011] Complete graphs. K2 K1 C3 C5

40. UpperDimension and ResolvingNumber dim+(G)= maximumsize of a minimalresolving set res(G)= minimumksuchthateveryk-subsetis a resolving set. (1,1,2,2,3) UPPER BASIS dim(G) ≤ dim+(G) ≤ res(G) (1,1,2,2,3) Realizability ??? dim+(G) = 4 res(G) = 6

41. Realizability [Chartrand et al.,2000] dim(G) ≤ dim+(G) ≤ res(G) = = a c dim(Kn)=n-1 res(Kn)=n-1

42. Realizability [Chartrand et al.,2000] dim(G) ≤ dim+(G) ≤ res(G) = = = b a c

43. Realizability [Chartrand et al.,2000] dim(G) ≤ dim+(G) ≤ res(G) = = b a Theorem: Conjecture:Foreverypaira,bof integerswith 2≤a≤b, thereexists a conectedgraphG suchthatdim(G)=a and dim+(G)=b. Itis true!!! [Garijo,G.,Márquez,2011]

44. Realizability [Chartrand et al.,2000] dim(G) ≤ dim+(G) ≤ res(G) = = b a Howmany???

45. Realizability [Chartrand et al.,2000] dim(G) ≤ dim+(G) ≤ res(G) = = b a Howmany???

46. Realizability [Chartrand et al.,2000] dim(G) ≤ dim+(G) ≤ res(G) = = b a Howmany???

47. Realizability [Chartrand et al.,2000] dim(G) ≤ dim+(G) ≤ res(G) = = b a Howmany???

48. Realizability [Chartrand et al.,2000] dim(G) ≤ dim+(G) ≤ res(G) = = = b a c Theorem:[Garijo,G.,Márquez] Given c>3, the set of graphswithresolvingnumbercisfinite. Howmany??? QUESTION (1): Realization of triples (a,b,c). QUESTION (2): RECONSTRUCTION!!!

49. Reconstruction Problem: givenc > 0, which are thegraphsG suchthat res(G) = c? res ≤ 2 [Chartrand,Zhang,2000] Paths and oddcycles.

50. Reconstruction Problem: givenc > 0, which are thegraphsG suchthat res(G) = c? res ≤ 2 [Chartrand,Zhang,2000] Paths and oddcycles. res = 3 [Garijo,G.,Márquez,2011] Evencycles plus other 18 graphs.