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Planar Graphs and Partially Ordered Sets

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Planar Graphs and Partially Ordered Sets

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    1. Planar Graphs and Partially Ordered Sets William T. Trotter Georgia Institute of Technology

    2. Inclusion Orders

    3. Incidence Posets

    4. Vertex-Edge Posets

    5. Vertex-Edge-Face Posets

    6. Vertex-Edge-Face Posets for Planar Graphs

    7. Triangle Orders

    8. Circle Orders

    9. N-gon Orders

    10. Dimension of Posets The dimension of a poset P is the least t so that P is the intersection of t linear orders. Alternately, dim(P) is the least t for which P is isomorphic to a subposet of Rt

    11. A 2-dimensional poset

    12. A 3-dimensional poset

    13. A Family of 3-dimensional Posets

    14. Standard Examples of n-dimensional posets

    15. Another Example of an n-dimensional Poset

    16. Complexity Issues It is easy to show that the question: dim(P) = 2? is in P. Yannakakis showed in 1982 that the question: dim(P) = t? is NP-complete for fixed t = 3. The question: dim(P) = t? is NP-complete for height 2 posets for fixed t = 4. Still not known whether: dim(P) = 3? is NP-complete for height 2 posets.

    17. Schnyder’s Theorem (1989)

    18. Proposition

    19. Schnyder’s Theorem (restated)

    20. 3-Connected Planar Graphs Theorem (Brightwell and Trotter, 1993): If G is a planar 3-connected graph and P is the vertex-edge-face poset of G, then dim(P) = 4. The removal of any vertex or any face from P reduces the dimension to 3.

    21. Convex Polytopes in R3

    22. Convex Polytopes in R3 Theorem (Brightwell and Trotter, 1993): If M is a convex polytope in R3 and P is its vertex-edge-face poset, then dim(P) = 4. The removal of any vertex or face from P reduces the dimension to 3.

    23. Planar Multigraphs

    24. Planar Multigraphs Theorem (Brightwell and Trotter, 1997): Let D be a non-crossing drawing of a planar multigraph G, and let P be the vertex-edge-face poset determined by D. Then dim(P) = 4. Different drawings may determine posets with different dimensions.

    25. The Kissing Coins Theorem

    26. Planar Graphs and Circle Orders

    27. Remarks on Circle Orders Every poset of dimension at most 2 is a circle order – in fact with circles having co-linear centers. Using Warren’s theorem and the Alon/Scheinerman degrees of freedom technique, it follows that “almost all” 4-dimensional posets are not circle orders.

    28. Standard Examples are Circle Orders

    29. More Remarks on Circle Orders Every 2-dimensional poset is a circle order. For each t = 3, some t-dimensional posets are circle orders. But, for each fixed t = 4, almost all t-dimensional posets are not circle orders. Every 3-dimensional poset is an ellipse order with parallel major axes.

    30. Fundamental Question for Circle Orders (1984)

    31. Support for a Yes Answer

    32. Support for a No Answer

    33. More Troubling News

    34. A Triumph for Ramsey Theory

    35. Schnyder’s Theorem

    36. Easy Direction (Babai and Duffus, 1981)

    37. Easy Direction

    38. The Proof of Schnyder’s Theorem Normal labelings of rooted planar triangulations. Uniform angle lemma. Explicit decomposition into 3 forests. Inclusion property Three auxiliary partial orders

    39. A Normal Labeling

    40. Normal Labeling - 1

    41. Normal Labeling - 2

    42. Normal Labeling - 3

    43. Lemma (Schnyder)

    44. Uniform Angles on a Cycle

    45. Uniform Angle Lemma (Schnyder)

    46. Suppose C has no Uniform 0

    47. Case 1: C has a Chord

    48. Uniform 0 on Top Part

    49. Uniform 0 on Bottom Part

    50. Faces Labeled Clockwise: Contradiction!!

    51. Case 2: C has No Chords

    52. Remove a Boundary Edge

    53. Without Loss of Generality

    54. Labeling Properties Imply:

    55. Remove Next Edge

    56. Continue Around Cycle

    57. The Contradiction

    58. Three Special Edges

    59. Shared Edges

    60. Local Definition of a Path

    61. Red Path from an Interior Vertex

    62. Red Path from an Interior Vertex

    63. Red Path from an Interior Vertex

    64. Red Path from an Interior Vertex

    65. Red Cycle of Interior Vertices??

    66. Red Path Ends at Exterior Vertex r0

    67. Red and Green Paths Intersect??

    68. Three Vertex Disjoint Paths

    69. Inclusion Property for Three Regions

    70. Explicit Partition into 3 Forests

    71. Final Steps The regions define three inclusion orders on the vertex set. Take three linear extensions. Insert the edges as low as possible. The resulting three linear extensions have the incidence poset as their intersection. Thus, dim(P) = 3.

    72. Grid Layouts of Planar Graphs

    73. Corollary (Schnyder, 1990)

    74. Algebraic Structure

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