Bijective Links on Planar Maps via Orientations

Transcription

1 Bijective Links on Planar Maps via Orientations Éric Fusy Dept. Math, University of British Columbia. p.1/1

2 Planar maps Planar map = graph drawn in the plane without edge-crossing, taken up to isotopy (continuous structure-preserving transformation) Rooted map = map + root edge rooting Motivations: mesh compression, graph drawing + nice combinatorial properties. p.2/26

3 Families of planar maps Planar map Loopless map Nonseparable map (no separating vertex) Triangulation Irreducible triangulation (no separating triangle). p.3/26

4 Families of planar maps separating vertex Planar map Loopless map Nonseparable map (no separating vertex) Triangulation Irreducible triangulation (no separating triangle). p.3/26

5 Families of planar maps separating vertex Planar map Loopless map Nonseparable map (no separating vertex) separating triangle Triangulation Irreducible triangulation (no separating triangle). p.3/26

6 Enumeration of planar maps Symbolic approach: Tutte, Brown Bijective approach: Cori, Schaeffer, Bouttier-Di Francesco-Guitter Planar maps Eulerian Loopless Nonseparable #(n edges) = 2 3n (2n)! (n+2)!n! #(n edges) = 3 2n 1 (2n)! (n+2)!n! #(n edges)= 2(4n+1)! (n+1)!(3n+2)! #(n edges)= 4(3n 3)! (n 1)!(2n)! 4-regular #(n vert.) = 2 3n (2n)! (n+2)!n! Bicubic #(2n vert.) = 3 2n 1 (2n)! (n+2)!n! Triangulations #(n+3 vert.)= 2(4n+1)! (n+1)!(3n+2)! Irreducible #(n+3 vert.)= 4(3n 3)! (n 1)!(2n)!. p.4/26

7 Enumeration of planar maps Symbolic approach: Tutte, Brown Bijective approach: Cori, Schaeffer, Bouttier-Di Francesco-Guitter Planar maps Eulerian Loopless Nonseparable #(n edges) = 2 3n (2n)! (n+2)!n! #(n edges) = 3 2n 1 (2n)! (n+2)!n! #(n edges)= 2(4n+1)! (n+1)!(3n+2)! #(n edges)= 4(3n 3)! (n 1)!(2n)! regular #(n vert.) = 2 3n (2n)! (n+2)!n! Bicubic #(2n vert.) = 3 2n 1 (2n)! (n+2)!n! Triangulations #(n+3 vert.)= 2(4n+1)! (n+1)!(3n+2)! Irreducible #(n+3 vert.)= 4(3n 3)! (n 1)!(2n)!. p.4/26

8 Enumeration of planar maps Symbolic approach: Tutte, Brown Bijective approach: Cori, Schaeffer, Bouttier-Di Francesco-Guitter Planar maps Eulerian Loopless Nonseparable #(n edges) = 2 3n (2n)! (n+2)!n! #(n edges) = 3 2n 1 (2n)! (n+2)!n! #(n edges)= 2(4n+1)! (n+1)!(3n+2)! #(n edges)= 4(3n 3)! (n 1)!(2n)! regular #(n vert.) = 2 3n (2n)! (n+2)!n! Bicubic #(2n vert.) = 3 2n 1 (2n)! (n+2)!n! Triangulations #(n+3 vert.)= 2(4n+1)! (n+1)!(3n+2)! Irreducible #(n+3 vert.)= 4(3n 3)! (n 1)!(2n)! 1, 2 well known bijections (Tutte) 3 recursive bijection (Wormald) This talk: new bijective construction for 3 first bijective construction for 4. p.4/26

9 Well known bijections 1 Radial mapping (Tutte) Planar maps n edges 4-regular n vertices 2 Trinity mapping (Tutte) Eulerian maps n edges Bicubic maps 2n vertices. p.5/26

10 Overview of the talk 1) Bijection nonseparable maps irreducible triang + new duality relation for bipolar orientations 2) Bijection loopless maps triangulations nonseparable components, irreducible components, 3) Applications to random generation and encoding. p.6/26

11 Bijection between nonseparable maps and irreducible triangulations. p.7/26

12 Bipolar orientations Bipolar orientation = acyclic orientation with a unique source and a unique sink t t s s. p.8/26

13 Bipolar orientations Bipolar orientation = acyclic orientation with a unique source and a unique sink t t s s A map admits a bipolar orientation iff there is no separating vertex (nonseparable). p.8/26

14 Transversal structures Transversal structure = partition of inner edges into a red and a blue bipolar orientations that are transversal (introduced by Xin He 93) N W E S. p.9/26

15 Transversal structures Transversal structure = partition of inner edges into a red and a blue bipolar orientations that are transversal (introduced by Xin He 93) N W E S A triangulation of the 4-gon admits a transversal structure iff there is no separating triangle (irreducible). p.9/26

16 Reformulating the bijection Nonseparable Irreducible?. p.10/26

17 Reformulating the bijection Nonseparable Irreducible? Ossona de Mendez 94 Fusy 05 No pattern No patterns Bipolar orientations Transversal structures. p.10/26

18 Reformulating the bijection Nonseparable Irreducible? Ossona de Mendez 94 Fusy 05 No pattern No patterns Bipolar orientations Transversal structures. p.10/26

19 How the bijection works Bipolar n edges Bipolar intransitive n + 1 vertices without Transversal structures n + 3 vertices without. p.11/26

20 How the bijection works Bipolar n edges Bipolar intransitive n + 1 vertices without Transversal structures n + 3 vertices without No pattern No pattern No patterns. p.11/26

21 Bipolar Bipolar intransitive Start with a plane bipolar orientation. p.12/26

22 Bipolar Bipolar intransitive Double the root edge. p.12/26

23 Bipolar Bipolar intransitive Insert a white vertex in each edge. p.12/26

24 Bipolar Bipolar intransitive Triangulate the faces by red edges. p.12/26

25 Bipolar Bipolar intransitive face edge additions. p.12/26

26 Bipolar Bipolar intransitive. p.12/26

27 Bipolar Bipolar intransitive A first bijection: Bipolar n edges Bipolar intransitive n + 3 vertices without 1) place a black vertex in each face 2) insert one black edge for each white vertex. p.12/26

28 Bipolar Bipolar intransitive A first bijection: Bipolar n edges Bipolar intransitive n + 3 vertices without 1) place a black vertex in each face 2) insert one black edge for each white vertex Remark: These are counted by the Baxter number: B n = 2 n(n+1) 2 n 1 ( n+1 k k=0 )( n+1 k+1 )( ) n+1 k+2. p.12/26

29 Bipolar intransitive Trans. struct. Start with an intransitive bipolar orientation. p.13/26

30 Bipolar intransitive Trans. struct. Triangulate the faces by blue edges. p.13/26

31 Bipolar intransitive Trans. struct. edge additions. p.13/26

32 Bipolar intransitive Trans. struct. A second bijection: Bipolar intransitive n + 1 vertices without Transversal structures n + 3 vertices without. p.13/26

33 The bijection Bipolar n edges Bipolar intransitive n + 1 vertices without Transversal structures n + 3 vertices without. p.14/26

34 The bijection Bipolar n edges Bipolar intransitive n + 1 vertices without Transversal structures n + 3 vertices without No pattern No pattern No patterns. p.14/26

35 The bijection Bipolar n edges No pattern Bipolar intransitive n + 1 vertices without No pattern Transversal structures n + 3 vertices without No patterns. p.14/26

36 The bijection Bipolar n edges No pattern Bipolar intransitive n + 1 vertices without No pattern Transversal structures n + 3 vertices without No patterns bijection Nonseparable n edges Irreducible n + 3 vertices. p.14/26

37 Bijection between loopless maps and triangulations. p.15/26

38 Decomposing a loopless map Block decomposition. p.16/26

39 Decomposing a loopless map Block decomposition For rooted loopless maps: Nonseparable core where each corner is possibly occupied by a loopless map. p.16/26

40 Decomposing a triangulation Classical decomposition at separating triangles. p.17/26

41 Decomposing a triangulation Classical decomposition at separating triangles 4-connected triangulation where each face is possibly occupied by a triangulation. p.17/26

42 Decomposing a triangulation Classical decomposition at separating triangles 4-connected triangulation where each face is possibly occupied by a triangulation Here: the same after deleting an outer edge. p.17/26

43 Decomposing a triangulation Classical decomposition at separating triangles 4-connected triangulation where each face is possibly occupied by a triangulation Here: the same after deleting an outer edge Irreducible triangulation where each face is possibly occupied by a triangulation. p.17/26

44 The decompositions are parallel A loopless map M ( M = # edges) is i) the vertex-map or ii) M 1 A triangulation T ( T = # vert. -3) is i) the triangle-map or ii) T 1 Nonseparable core C 2 C loopless maps M 1,..., M 2 C,,,,,,, M = C + 2 C i=1 M i Irreducible core I 2 I triangulations T 1,..., T 2 I,,,,,,, T = I + 2 I i=1 T i. p.18/26

45 The decompositions are parallel A loopless map M ( M = # edges) is i) the vertex-map or ii) M 1 A triangulation T ( T = # vert. -3) is i) the triangle-map or ii) T 1 Nonseparable core C 2 C loopless maps M 1,..., M 2 C,,,,,,, M = C + 2 C i=1 M i Irreducible core I 2 I triangulations T 1,..., T 2 I,,,,,,, T = I + 2 I i=1 T i Nonseparable maps n edges Loopless maps n edges bijection bijection Irreducible triang. n + 3 vertices Triangulations n + 3 vertices. p.19/26

46 Results obtained so far rooted nonseparable maps with n edges bipolar orientations rooted irreducible triangulations with n+3 vertices rooted loopless maps with n edges parallel decompositions rooted triangulations with n+3 vertices. p.20/26

47 Results obtained so far rooted nonseparable maps with n edges bipolar orientations rooted irreducible triangulations with n+3 vertices rooted loopless maps with n edges parallel decompositions rooted triangulations with n+3 vertices N n = I n L n = T n. p.20/26

48 Results obtained so far rooted nonseparable maps with n edges bipolar orientations rooted irreducible triangulations with n+3 vertices rooted loopless maps with n edges parallel decompositions rooted triangulations with n+3 vertices N n = I n =? L n = T n =?. p.20/26

49 Counting the families. p.21/26

50 Irreducible triang. ternary trees Fusy 05: irreductible triangulations are in bijection with ternary trees. p.22/26

51 Irreducible triang. ternary trees canonical transversal structure Fusy 05: irreductible triangulations are in bijection with ternary trees. p.22/26

52 Irreducible triang. ternary trees canonical transversal structure Fusy 05: irreductible triangulations are in bijection with ternary trees. p.22/26

53 Irreducible triang. ternary trees canonical transversal structure Fusy 05: irreductible triangulations are in bijection with ternary trees rooting. p.22/26

54 Triangulations quaternary trees Poulalhon-Schaeffer 03, Fusy-Poulalhon-Schaeffer 05: triangulations are in bijection with quaternary trees. p.23/26

55 Triangulations quaternary trees Poulalhon-Schaeffer 03, Fusy-Poulalhon-Schaeffer 05: triangulations are in bijection with quaternary trees. p.23/26

56 Triangulations quaternary trees Poulalhon-Schaeffer 03, Fusy-Poulalhon-Schaeffer 05: triangulations are in bijection with quaternary trees. p.23/26

57 Triangulations quaternary trees Poulalhon-Schaeffer 03, Fusy-Poulalhon-Schaeffer 05: triangulations are in bijection with quaternary trees. p.23/26

58 Triangulations quaternary trees Poulalhon-Schaeffer 03, Fusy-Poulalhon-Schaeffer 05: triangulations are in bijection with quaternary trees. p.23/26

59 Triangulations quaternary trees Poulalhon-Schaeffer 03, Fusy-Poulalhon-Schaeffer 05: triangulations are in bijection with quaternary trees. p.23/26

60 Enumerative Results rooted nonseparable maps with n edges rooted loopless maps with n edges rooted irreducible triangulations with n+3 vertices rooted triangulations with n+3 vertices rooted ternary trees with n 1 nodes rooted quaternary trees with n nodes. p.24/26

61 Enumerative Results rooted nonseparable maps with n edges rooted loopless maps with n edges rooted irreducible triangulations with n+3 vertices rooted triangulations with n+3 vertices rooted ternary trees with n 1 nodes rooted quaternary trees with n nodes N n = T n = 4(3n 3)! (n 1)!(2n)! L n = I n = 2(4n+1)! (n+1)!(3n+2)!. p.24/26

62 Enumerative Results rooted nonseparable maps with n edges rooted loopless maps with n edges Schaeffer 99 rooted irreducible triangulations with n+3 vertices rooted triangulations with n+3 vertices? rooted ternary trees with n 1 nodes rooted quaternary trees with n nodes N n = T n = 4(3n 3)! (n 1)!(2n)! L n = I n = 2(4n+1)! (n+1)!(3n+2)!. p.24/26

63 Applications. p.25/26

64 Algorithmic applications General scheme: (Schaeffer 99, Poulalhon-Schaeffer 03) maps trees Dyck words. p.26/26

65 Algorithmic applications General scheme: (Schaeffer 99, Poulalhon-Schaeffer 03) maps trees Dyck words optimal encoding (mesh compression). p.26/26

66 Algorithmic applications General scheme: (Schaeffer 99, Poulalhon-Schaeffer 03) maps trees Dyck words optimal encoding (mesh compression) random generation (random discrete surfaces). p.26/26

67 Algorithmic applications General scheme: (Schaeffer 99, Poulalhon-Schaeffer 03) maps trees Dyck words optimal encoding (mesh compression) random generation (random discrete surfaces) Applies here to : irreducible triangulations triangulations. p.26/26

68 Algorithmic applications General scheme: (Schaeffer 99, Poulalhon-Schaeffer 03) maps trees Dyck words optimal encoding (mesh compression) random generation (random discrete surfaces) Applies here to : as well as : irreducible triangulations triangulations loopless maps nonseparable maps. p.26/26