FSP-Seminar

1. FSP-Seminar March 2007, Graz Maximizing Maximal Anglesfor Plane Straight-Line Graphs O. Aichholzer, T. Hackl, M. Hoffmann, C. Huemer, A. Pór, F. Santos, B. Speckmann, B. Vogtenhuber Graz University of Technology, Austria ETH Zürich, SwitzerlandUniversitat Politècnica de Catalunya, Spain Hungarian Academy of Sciences, Hungary Universidad de Cantabria, Spain TU Eindhoven, Netherlands

2. Plane Geometric Graphs • vertices: • n points in the plane • points in general position • edges: • straight lines spanned by vertices (geometric graphs) • no crossings (plane) 1

3. Plane Geometric Graphs • perfect matchings 2

4. Plane Geometric Graphs • perfect matchings • spanning paths 2

5. Plane Geometric Graphs • perfect matchings • spanning paths • spanning trees 2

6. Plane Geometric Graphs • perfect matchings • spanning paths • spanning trees • connected plane graphs 2

7. Plane Geometric Graphs • perfect matchings • spanning paths • spanning trees • connected plane graphs • spanning cycles 2

8. Plane Geometric Graphs • perfect matchings • spanning paths • spanning trees • connected plane graphs • spanning cycles • triangulations 2

9. Plane Geometric Graphs • perfect matchings • spanning paths • spanning trees • connected plane graphs • spanning cycles • triangulations • pseudo-triangulations 2

10. Basic Idea Generalizing the principle of large incident angles of pointed pseudo-triangulations to other classes of plane graphs 3

11. Pseudo-Triangulations • pseudo-triangle • 3 convex vertices • concave chains 4

12. Pseudo-Triangulations • pseudo-triangle • 3 convex vertices • concave chains • pseudo-triangulation • convex hull • partitioned intopseudo-triangles • pointed: each point has an incident angle of at least p 4

13. p q a - Openness point set S, graph G(S) • A point in p  S is a – open in G(S), if it has an incident angle of at least a • The graph G is a – open, if every point in S is a – open in G(S) • A class Gof graphs is a – open, if for all point sets S there exists an a – open graph G(S) of class G() a 5

14. The Question We know that pointed pseudo-triangulations are p – open. Can we generalize this concept to other classes of graphs? Given a class G(of graphs, • Does there exist some angle a, such that G(is a – open? • If yes, what is the maximal such a? 6

15. Min Max Min Max problem • Optimization for class G of plane graphs: • true for all sets S, even for the worst • for S: take the best graph G(S)G • has to hold for any point p in G(S) • for a point p take the maximum incident angle • find maximal  for each class:minS maxGG minpS maxaA(p,G){a} 7

16. Triangulations • convex hull points are p – open 8

17. Triangulations • convex hull points are p – open • take the convex hull • triangulate 8

18. Triangulations • triangular convex hull (edges a,b,c) • closest point for each edge (a‘,b‘,c‘) • hexagon with hull points and closest edge points • triangles empty • one angle {a,b,g} ≥2p/3 • choose • connect • recurse on smaller subproblems b a b‘ a b a‘ c‘ g c 9

19. Triangulations Theorem 1: Triangulations are 2p/3-open. Moreover, this bound is best possible. 10

20. Spanning Trees • not more than 5p/3-open: • at least 3p/2-open: • at least 5p/3-open: 11

21. Spanning Trees • Not more than 5p/3-open: • At least 3p/2-open: • At least 5p/3-open: • diameter • farthest points • case analysis on angles 11

22. Spanning Trees • not more than 5p/3-open: • at least 3p/2-open: • at least 5p/3-open: • diameter • farthest points • case analysis on angles Theorem 2: (general) Spanning Trees are 5p/3-open, and this bound is best possible. 11

23. Spanning Trees (bounded vertex degree 3) • At least 3p/2-open: • start with diameter • assign subsets • recursively take diameters • consider tangents • connect subsets Theorem 3: Spanning Trees with maximum vertex degree of at most 3 are 3p/2-open. 12

24. Spanning Trees (bounded vertex degree 3) 3p/2 Theorem 3: Spanning Trees with maximum vertex degree of at most 3 are 3p/2-open. Moreover, this bound is best possible. 12

25. Spanning Trees (bounded vertex degree 3) 3p/2 Corrolary: Connected Graphs with bounded vertex degree of at most n-2 are at most 3p/2-open. Theorem 3: Spanning Trees with maximum vertex degree of at most 3 are 3p/2-open. Moreover, this bound is best possible. 12

26. Spanning Paths (convex point sets) • inner angles (consecutive points): • at most one angle  p/2 • diameter points: no angle  p/2 • in total ≤ (n-2) angles  p/2 • „zig-zag“ spanning paths: • two paths per point • each path counted twice • in total nzig-zag paths + each inner angle occurs in exactly one zig-zag path  at least two zig-zag paths with no angle  p/2 <p Theorem 4: Spanning Paths (for convex sets) are 3p/2-open, and this bound is best possible. 13

27. Spanning Paths (general) Theorem 4: Spanning Paths are 5p/4-open. • For every vertex q of the convex hull of S, there exists a 5p/4-open spanning path on S starting at q. • For every edge q1q2 of the convex hull of S, there exists a 5p/4-open spanning path on S starting with q1q2. • Case analysis over occuring angles • Proof by induction over the number of points,(1) and (2) not independent 14

28. ??? Conclusion   • Pointed Pseudo-Triangulations p (180°) • Perfect Matchings 2p (360°) • Spanning Cycles p (180°) • Triangulations 2p/3 (120°) • Spanning Trees (unbounded) 5p/3 (300°) • Spanning Trees with bounded vertex degree 3p/2 (270°) • Spanning Paths (convex) 3p/2 (270°) • Spanning Paths (general) 5p/4(225°)      5p/4 (225°) – 3p/2 (270°) 15

29. Thanks! Thanks for your attention … Danke Gracias Efcharisto Dank U wel Grazie Merci 16