Birgit Vogtenhuber

On the number of plane graphs

We investigate the number of plane geometric, i.e., straight-line, graphs, a set <i>S</i> of <i>n</i> points in the plane admits. We show that the number of plane graphs and connected plane graphs as well as the number of cycle-free plane graphs is minimized when <i>S</i> is in convex position. Moreover, these results hold for all these graphs with an arbitrary but fixed number of edges. Consequently, we provide simple proofs that the number of spanning trees, cycle-free graphs (forests), perfect matchings, and spanning paths is also minimized for point sets in convex position.In addition we construct a new extremal configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ*(√72^<i>n</i>) = Θ*(8.4853^<i>n</i>) triangulations and Θ*(41.1889^<i>n</i>) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples.

Evacuating Robots from a Disk Using Face-to-Face Communication Extended Abstract

Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed

Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons

1

Pointed drawings of planar graphs

. The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [9] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most

Matching edges and faces in polygonal partitions

5.740

On k-convex point sets

and also proved that any algorithm has evacuation time at least