2. Extremal Configurations and Levels in Pseudoline Arrangements Shakhar Smorodinsky (Tel-Aviv University) Joint work with Micha Sharir

3. 2 What the … am I going to show/talk about? n Points are like n Lines, which are almost like n Pseudolines

4. 3 What the … am I going to show/talk about? n So sometimes pseudolines are like points

5. 4 Geometric graphs n G = (V,E) simple graph + an embedding n Consider crossing edges n What does it mean in a ‘dual’ setting? n Well, … what does ‘dual’ mean?

6. 5  duality between pts and lines n point p=(a,b) => line p * : y = -ax+b n line l: y=cx+d => point l * =(c,d) n Preserving the above/below relationship l p p*p* l*l*

7. 6 Geometric graphs and duality n Consider a crossing edge pair Lets dualize as follows: a b c d a*a* b*b* c*c* d*d* A diamond

8. 7 Geometric graphs and duality n Consider a collection  of n lines plus vertices E with NO diamonds. G=( , E )

9. 8 Geometric graphs and duality (cont) n This is a graph which can be drawn in the plane with no crossings. Hence planar graph!

10. 9 Geometric graphs and duality (cont) n Does this also hold for pseudolines? n Yes! [Tamaki Tokuyama 97]

11. 10 Geometric graphs and duality (cont) n Thm: [Tamaki Tokuyama 97] A diamond-free Pseudoline graph is planar. Proof: complicated. We provide a very simple proof.

12. 11 Geometric graphs and duality (cont) n A diamond-free Pseudoline graph is planar. n Simple proof (sketch): We draw the graph as follows:

13. 12 Diamond free pseudolines graph (cont) n in this drawing any pair of edges intersect an even # times. Using Tutte [70] implies planarity

14. 13 Pseudolines and Diamonds (cont) n What is diamond good for? n It is an auxiliary structure that helps analyzing the complexity of the k-level. n k-level is the set of all vertices that lie above exactly k pseudolines

15. 14 Pseudolines and Diamonds (cont) The 2-level

16. 15 k-level (cont) What is the max complexity of a k-level in n (pseudo) lines? Major open problem in combinatorial geometry! Known upper bound: O(nk 1/3 ) [Dey 98 (for lines)] [Tamaki Tokuyama 97 (for pseudolines)] Both proofs uses the notion of diamond.

17. 16 More on Pseudolines n We extend many results on geometric graphs to the more general case of pseudolines graphs n Example: A Thrackle is a graph drawn in the plane such that any pair of edges intersect.

18. 17 Thrackles A Thrackle is a graph drawn in the plane such that any pair of edges intersect. n Conjecture (Conway): The max # edges of a Thrackle on n pts is n. n Known for drawings with straight line segments. [Perles] n For general drawings the best upper bound is 2n-3 [Cairns, Nikolayevsky 2000]

19. 18 Thrackles (cont) n We prove the conjecture for thrackles drawn with ‘extendible pseudo-segments’, by showing: n If, in a set of vertices in n pseudolines each pair form a diamond then #vertices is at most n

20. 19 More on Pseudolines n Consider a geometric graph: n Parallel edges n Thm: [Katchalski Last 98], [Valtr 98] Max # edges of geometric graph with no parallel edges is 2n-2

21. 20 Geometric graphs with no parallel edges n Thm: [Katchalski Last 98], [Valtr 98] Max # edges of geometric graph with no parallel edges is 2n-2

22. 21 Geometric graphs with no parallel edges n We generalize this result to the context of pseudolines graph. a b c d b*b* a*a* c*c* d*d* anti-diamond n What is the dual of parallel edges ?

23. 22 Pseudolines and anti-diamonds n Thm [S, Sharir]: An anti-diamond free pseudolines graph contains at most 2n-2 ‘edges’ a b c d b*b* a*a* c*c* d*d* anti-diamond

24. 23 THANK YOU Wake up!!!