Birgit Vogtenhuber

Håvard Bakke Bjerkevik, Joseph Dorfer, Linda Kleist et al.

We consider the problem of reconfiguring non-crossing spanning trees on point sets. For a set $P$ of $n$ points in general position in the plane, the flip graph $F(P)$ has a vertex for each non-crossing spanning tree on $P$ and an edge between any two spanning trees that can be transformed into each other by the exchange of a single edge. This flip graph has been intensively studied, lately with an emphasis on determining its diameter diam$(F(P))$ for sets $P$ of $n$ points in convex position. The current best bounds are $\frac{14}{9}n-O(1) \leq$ diam$(F(P))<\frac{15}{9}n-3$ [Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber; SODA 2025]. The crucial tool for both the upper and lower bound are so-called *conflict graphs*, which the authors stated might be the key ingredient for determining the diameter (up to lower-order terms). In this paper, we pick up the concept of conflict graphs and show that this tool is even more versatile than previously hoped. As our first main result, we use conflict graphs to show that computing the flip distance between two non-crossing spanning trees is NP-hard, even for point sets in convex position. Interestingly, the result still holds for more constrained flip operations, concretely, compatible flips (where the removed and the added edge do not cross) and rotations (where the removed and the added edge share an endpoint). Extending the line of research from [BKUV SODA25], we present new insights on the diameter of the flip graph. Their lower bound is based on a constant-size pair of trees, one of which is *stacked*. We show that if one of the trees is stacked, then the lower bound is indeed optimal up to a constant term, that is, there exists a flip sequence of length at most $\frac{14}{9}(n-1)$ to any other tree. Lastly, we improve the lower bound on the diameter of the flip graph $F(P)$ for $n$ points in convex position to $\frac{11}{7}n-o(n)$.

Panos Giannopoulos, Miriam Goetze, Grzegorz Gutowski et al.

Drawing graphs with the minimum number of crossings is a classical problem that has been studied extensively. Many restricted versions of the problem have been considered. For example, bipartite graphs can be drawn such that the two sets in the bipartition of the vertex set are mapped to two parallel lines, and the edges are drawn as straight-line segments. In this setting, the number of crossings depends only on the ordering of the vertices on the two lines. Two natural variants of the problem have been studied. In the one-sided case, the order of the vertices on one of the two lines is given and fixed; in the two-sided case, no order is given. Both cases are important yet NP-hard subproblems in the so-called Sugiyama framework for drawing layered graphs with few crossings. For the one-sided case, Eades and Wormald [Algorithmica 1994] introduced a median heuristic and showed that it has an approximation ratio of 3. In recent years, researchers have focused on a local version of crossing minimization, where the aim is to minimize the maximum number of crossings per edge instead of the total number of crossings. Kobayashi, Okada, and Wolff [SoCG 2025] investigated the complexity of local crossing minimization parameterized by the natural parameter. They conjectured that one-sided local crossing minimization is NP-hard. In this work, we confirm their conjecture by showing that the problem is NP-hard even for forests of high-degree stars. In fact, more strongly, the reduction yields a tight lower bound, which excludes the existence of subexponential-time algorithms assuming the Exponential-Time Hypothesis. In contrast, we present a quadratic-time algorithm for the special case of forests of stars of maximum degree 2. Finally, we provide a median heuristic with a carefully designed tie-breaking scheme and prove that it has an approximation ratio of 3 in the local setting.

Daniel Bertschinger, Henry Förster, Fabian Klute et al.

We study the common intersection of arrangements of double-wedges. We consider arrangements where double-wedges may be either bowties (which do not contain a vertical line) or hourglasses (which contain a vertical line), in contrast to earlier studies that focused on arrangements of only bowties. This generalization changes the setting drastically, in particular, with respect to all arguments involving the point-line duality. Namely, a point in the intersection of all double-wedges is equivalent to a line that stabs a set of segments $\mathcal{S}$ (corresponding to the bowties) while it avoids a different set of segments $\mathcal{A}$ (corresponding to the complement of the hourglasses). We show that in this general setting, the intersection of $n$ double-wedges may consist of $Ω(n^2)$ interior-disjoint regions. Further, we discuss Gallai-type results for arrangements of segments and anti-segments, and we provide algorithms for computing the intersection of such arrangements with worst-case optimal running time. Finally, we also prove that we can find a single intersection point in almost optimal running time, assuming that 3SUM admits no truly subquadratic-time algorithm.

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}.