# Best Paper Award for Felix Schröder

Dr. Felix Schröder, a postdoctoral researcher at the Department of Applied Mathematics at Charles University, received the award for the best paper at the Graph Drawing and Network Visualization Conference GD‘24.

The annual international symposium Graph Drawing and Network
Visualization focuses on the combinatorial and algorithmic aspects of graph
drawing and the design of systems and interfaces for network visualization. At
the 32^{nd} symposium, held from September 18 to 20 in Vienna, Dr.
Schröder presented a paper titled *“The Density Formula: One Lemma to
Bound them All”*. For this article, he also received the *Best Paper
Award*.

Dr. Schröder's research concerns with graph theory and
combinatorial geometry. In the award-winning paper, he, along with colleagues,
presents a new lemma (auxiliary statement) useful for graph drawing in and out
of plane. *“Our lemma can be used to prove bounds on the so-called density
of non-planar graphs. In this context, graphs are networks where some nodes (or
vertices) are connected by edges. The number of edges in various graphs is
referred to as ‚density.‘ Density is an important topic because if you can
prove that there aren't too many edges, the graph is considered sparse, which
often means that algorithms can be run efficiently on it,”* explains Dr.
Schröder.

In the context of graph drawing, the most well-known class is
planar graphs. These are graphs where vertices can be drawn as points in
two-dimensional space and all edges as line segments between these points such
that no two edges intersect— the only common point, they may have, is at the
endpoints. However, this is not entirely true for classes beyond planar graphs.
*“For example, in k-planar graphs, edges may cross, but only a few times
(at most k times), while in so-called rectilinear (RAC) graphs, crossings can
only occur at right angles. In our article, we proposed and proved a new
lemma—derived almost directly from the famous Euler's formula—and used it
to provide a uniform proof of many results regarding the density of graph
classes close to planar graphs that were already known in the literature. With
the help of this lemma, we answered several open questions, such as the density
of RAC graphs. Our proof technique is so simple that we were able to include
many results in the article, all of which now have simpler proofs than before
(if they had any at all),”* explains Dr. Schröder.

Dr. Schröder
joined the Department of Applied Mathematics last November after defending his
doctoral thesis at the Technical University of Berlin. *“There are many
excellent scientists working at Matfyz, some of whom I had already collaborated
with, so I knew that moving to Prague was a good decision,”* Dr.
Schröder responds when asked why he chose to undertake a fellowship at Charles
University in the DiGeo group led by Associate Professor Pavel Valtr. His
contract at the faculty ends at the end of the year. *“I will apply for a one-month extension; I am not clear about my further plans, but, if possible, I would like to stay close to Berlin, where most of my family and friends
live,”* the young scientist outlines.