In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs). Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars. Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

This paper proposes a computationally efficient method to estimate the time-varying relative pose between two visual-inertial sensor rigs mounted on the flexible wings of a fixed-wing unmanned aerial vehicle (UAV). The estimated relative poses are used to generate highly accurate depth maps in real-time and can be employed for obstacle avoidance in low-altitude flights or landing maneuvers. The approach is structured as follows: Initially, a wing model is identified by fitting a probability density function to measured deviations from the nominal relative baseline transformation. At runtime, the prior knowledge about the wing model is fused in an Extended Kalman filter (EKF) together with relative pose measurements obtained from solving a relative perspective N-point problem (PNP), and the linear accelerations and angular velocities measured by the two inertial measurement units (IMU) which are rigidly attached to the cameras. Results obtained from extensive synthetic experiments demonstrate that our proposed framework is able to estimate highly accurate baseline transformations and depth maps.