Balanced Schnyder woods for planar triangulations: an experimental study with applications to graph drawing and graph separators

In this work we consider balanced Schnyder woods for planar graphs, which are Schnyder woods where the number of incoming edges of each color at each vertex is balanced as much as possible. We provide a simple linear-time heuristic leading to obtain well balanced Schnyder woods in practice. As test applications we consider two important algorithmic problems: the computation of Schnyder drawings and of small cycle separators. While not being able to provide theoretical guarantees, our experimental results (on a wide collection of planar graphs) suggest that the use of balanced Schnyder woods leads to an improvement of the quality of the layout of Schnyder drawings, and provides an efficient tool for computing short and balanced cycle separators.Comment: Appears in the Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019

Attosecond hole migration in benzene molecules surviving nuclear motion

Hole migration is a fascinating process driven by electron correlation, in which purely electronic dynamics occur on a very short time scale in complex ionized molecules, prior to the onset of nuclear motion. However, it is expected that due to coupling to the nuclear dynamics, these oscillations will be rapidly damped and smeared out, which makes experimental observation of the hole migration process rather difficult. In this Letter, we demonstrate that the instantaneous ionization of benzene molecules initiates an ultrafast hole migration characterized by a periodic breathing of the hole density between the carbon ring and surrounding hydrogen atoms on a subfemtosecond time scale. We show that these oscillations survive the dephasing introduced by the nuclear motion for a long enough time to allow their observation. We argue that this offers an ideal benchmark for studying the influence of hole migration on molecular reactivity