2. Shakhar Smorodinsky ETH Zurich Joint work with Rom Pinchasi, MIT Locally Delaunay Geometric Graphs

3. Geometric Graphs A Geometric Graph is: G=(V,E) +embedding in R 2

4. K-locally Delaunay graphs A Geometric graph is k-locally Delaunay if: Can be embedded s.t. every edge is isolated from its (k)-neighbors by some disc 1-locally Delaunay

5. Problem 1: What is the maximum edge complexity of a 1-locally delaunay graph G=(V,E)? Motivation? Topology Control for Sensor Networks First observation: [S. Kapoor, X.Y. Li 03]: G cannot contain a K 3,3. Hence |E| = O(n 5/3 )

6. G contains no K 3,3. Hence |E| = O(n 5/3 ) [Kovari, Sos, Turan] Proof: As a matter of fact: if G can be embeded without a self-crossing C 4 then |E| = O(n 8/5 ). [Pinchasi, Radoicic 03].

7. Next improvement: Can G contain a K 2,2 ? Yes!

8. Our contribution: Thm [Pinchasi, S]: If G=(V,E) is 1-locally Delaunay then |E|=O(n 3/2 ) Lets assume many things: 1)All edges makes a small angle with the vector (0,1). 2) All edges cross the x-axis 3) For every edge e, the witnessing disc is such that its center is left to e

9. If G=(V,E) is 1-locally Delaunay then |E|=O(n 3/2 ) Proof:

10. If G=(V,E) is 1-locally Delaunay then |E|=O(n 3/2 ) Proof: (cont) Under these assumptions: G contains no K 2,2

11. If G=(V,E) is 1-locally Delaunay then |E|=O(n 3/2 ) Proof: (cont) Assume G contains K 2,2 : A contradiction

12. Thm [Pinchasi, S]: If G=(V,E) is 2- locally Delaunay then |E|=O(n) Remark: First observation: G contains no self crossing copy of P 3 Hence by [Pach, Pinchasi, Tardos, Toth 03] |E| = O(n log n)

13. If G=(V,E) is 2-locally Delaunay then |E|=O(n) Proof: Lets assume small angles between edges Remove the (upper, lower) right most and left most edge from every vertex

14. If G=(V,E) is 2-locally Delaunay then |E|=O(n) Proof: (cont) Claim: No edge survived!!!

15. Claim: No edge survived!!!