Birgit Vogtenhuber

Oswin Aichholzer, Joachim Orthaber, Birgit Vogtenhuber

Generalizing pseudospherical drawings, we introduce a new class of simple drawings, which we call separable drawings. In a separable drawing, every edge can be closed to a simple curve that intersects each other edge at most once. For different edges, the non-edge parts of these curves may interact arbitrarily though. Most notably, we show that (1) every separable drawing of any graph on $n$ vertices in the plane can be extended to a simple drawing of the complete graph $K_n$, (2) every separable drawing of $K_n$ contains a crossing-free Hamiltonian cycle and is plane Hamiltonian connected (that is, it contains a crossing-free Hamiltonian path between each pair of vertices), and (3) every generalized convex drawing and every 2-page book drawing is separable. Further, the class of separable drawings is a proper superclass of the union of generalized convex and 2-page book drawings. Hence, our results on plane Hamiltonicity extend recent work on generalized convex drawings by Bergold et al. (DCG 2025).

Jared Coleman, Evangelos Kranakis, Oscar Morales-Ponce et al.

In the pattern formation problem, robots in a system must self-coordinate to form a given pattern, regardless of translation, rotation, uniform-scaling, and/or reflection. In other words, a valid final configuration of the system is a formation that is \textit{similar} to the desired pattern. While there has been no shortage of research in the pattern formation problem under a variety of assumptions, models, and contexts, we consider the additional constraint that the maximum distance traveled among all robots in the system is minimum. Existing work in pattern formation and closely related problems are typically application-specific or not concerned with optimality (but rather feasibility). We show the necessary conditions any optimal solution must satisfy and present a solution for systems of three robots. Our work also led to an interesting result that has applications beyond pattern formation. Namely, a metric for comparing two triangles where a distance of $0$ indicates the triangles are similar, and $1$ indicates they are \emph{fully dissimilar}.