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Graph isomorphism and asymmetric graphs Pascal Schweitzer Ghent Graph Theory Workshop 2017 August 18th, Ghent Graph isomorphism and asymmetric graphs Pascal Schweitzer 1 / 38 asymmetry oracle graph isomorphism non-trivial automorphisms

  1. Tournaments A tournament is an oriented complete graph. (exactly one directed edge between every pair of vertices) Graph isomorphism and asymmetric graphs Pascal Schweitzer 17 / 38

  2. Tournaments A tournament is an oriented complete graph. (exactly one directed edge between every pair of vertices) User:Nojhan/Wikimedia Commons/CC-BY-SA-3.0 Graph isomorphism and asymmetric graphs Pascal Schweitzer 17 / 38

  3. Symmetry problems for tournaments GI Tour col-GI Tour Aut ( T ) | Aut ( T ) | Graph isomorphism and asymmetric graphs Pascal Schweitzer 18 / 38

  4. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  5. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  6. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  7. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  8. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  9. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  10. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  11. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  12. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  13. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  14. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  15. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  16. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  17. Removing colors for tournaments - colored tournament isomorphism � tournament isomorphism col-GI Tour ≤ p m GI Tour [Arvind, Das, Mukhopadhyay] (2010) - colored tournament asymmetry � tournament asymmetry col-GA Tour ≤ p m GA Tour Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

  18. Alternative to disjoint union for tournaments For tournaments we cannot form the disjoint union . Instead we form the triangle tournament Tri ( T 1 , T 2 ) . Graph isomorphism and asymmetric graphs Pascal Schweitzer 20 / 38

  19. Alternative to disjoint union for tournaments For tournaments we cannot form the disjoint union . Instead we form the triangle tournament Tri ( T 1 , T 2 ) . T 1 T 2 Graph isomorphism and asymmetric graphs Pascal Schweitzer 20 / 38

  20. Alternative to disjoint union for tournaments For tournaments we cannot form the disjoint union . Instead we form the triangle tournament Tri ( T 1 , T 2 ) . T 1 T 2 T 1 ∼ T ′ = T ′ 1 1 Graph isomorphism and asymmetric graphs Pascal Schweitzer 20 / 38

  21. Alternative to disjoint union for tournaments For tournaments we cannot form the disjoint union . Instead we form the triangle tournament Tri ( T 1 , T 2 ) . T 1 T 2 T 1 ∼ T ′ = T ′ 1 1 Graph isomorphism and asymmetric graphs Pascal Schweitzer 20 / 38

  22. Alternative to disjoint union for tournaments For tournaments we cannot form the disjoint union . Instead we form the triangle tournament Tri ( T 1 , T 2 ) . T 1 T 2 T 1 ∼ T ′ = T ′ 1 1 3 · | Aut ( T 1 ) | 2 · | Aut ( T 2 ) | � if T 1 ∼ = T 2 | Aut ( Tri ( T 1 , T 2 )) | = | Aut ( T 1 ) | 2 · | Aut ( T 2 ) | otherwise. Graph isomorphism and asymmetric graphs Pascal Schweitzer 20 / 38

  23. Asymmetry vs isomorphism for tournaments GI Tour col-GI Tour Aut ( T ) | Aut ( T ) | Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

  24. Asymmetry vs isomorphism for tournaments GI Tour col-GI Tour Aut ( T ) | Aut ( T ) | GI AsymTour GA Tour Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

  25. Asymmetry vs isomorphism for tournaments GI Tour col-GI Tour Aut ( T ) | Aut ( T ) | GI AsymTour GA Tour Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

  26. Asymmetry vs isomorphism for tournaments GI Tour col-GI Tour Aut ( T ) | Aut ( T ) | GI AsymTour GA Tour Open question: Is it harder to find all symmetries than to detect asymmetry? Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

  27. Asymmetry vs isomorphism for tournaments GI Tour col-GI Tour Aut ( T ) | Aut ( T ) | ? GI AsymTour GA Tour Open question: Is it harder to find all symmetries than to detect asymmetry? Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

  28. Asymmetry vs isomorphism for tournaments GI Tour col-GI Tour Aut ( T ) | Aut ( T ) | randomized GI AsymTour GA Tour Open question: Is it harder to find all symmetries than to detect asymmetry? Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

  29. Main Result Theorem There is a polynomial-time randomized reduction from tournament isomorphism to tournament asymmetry. Thus: For tournaments finding all symmetries and detecting asymmetry are polynomially equivalent. Graph isomorphism and asymmetric graphs Pascal Schweitzer 22 / 38

  30. asymmetry oracle graph isomorphism non-trivial automorphisms asymmetry invariant suborbits automorphism tournaments group and isomorphism

  31. asymmetry oracle graph isomorphism non-trivial automorphisms asymmetry invariant suborbits automorphism tournaments group and isomorphism

  32. asymmetry oracle graph isomorphism non-trivial automorphisms asymmetry invariant suborbits automorphism tournaments group and isomorphism

  33. asymmetry oracle graph isomorphism non-trivial automorphisms asymmetry invariant suborbits automorphism tournaments group and isomorphism

  34. Sampling automorphisms Technique 1: asymmetry test � non-trivial automorphism sampler Graph isomorphism and asymmetric graphs Pascal Schweitzer 25 / 38

  35. Sampling automorphisms Technique 1: asymmetry test � non-trivial automorphism sampler Strategy - fix more and more vertices until graph is asymmetric - make a copy of the graph - undo last fixing in copy - find alternative vertex to the vertex fixed last - find isomorphism from original to copy Graph isomorphism and asymmetric graphs Pascal Schweitzer 25 / 38

  36. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  37. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  38. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  39. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  40. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  41. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  42. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  43. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  44. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  45. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  46. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  47. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  48. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  49. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  50. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  51. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  52. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  53. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

  54. How to get automorphisms — Illustration Automorphisms: Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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