We study the common intersection of arrangements of double-wedges. We consider arrangements where double-wedges may be both 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 A (corresponding to the bowties) while it avoids a different set of segments A (corresponding to the complement of the hourglasses). We show that in this general setting, the intersection of n double-wedges may consist of Omega(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.

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 $\frac{11}{7}n-o(n)$.

We study the reconfiguration of plane spanning trees on point sets in the plane in convex position, where a reconfiguration step (flip) replaces one edge with another, yielding again a plane spanning tree. The flip distance between two trees is then the minimum number of flips needed to transform one tree into the other. We study structural properties of shortest flip sequences. The folklore happy edge conjecture suggests that any edge shared by both the initial and target tree is never flipped in a shortest flip sequence. The more recent parking edge conjecture, which would have implied the happy edge conjecture, states that there exist shortest flip sequences which use only edges of the start and target tree, and edges in the convex hull of the point set. Finally, another conjecture that is implicit in the literature is the reparking conjecture which states that no edge is flipped more than twice. Essentially all recent flip algorithms respect these three conjectures and the properties they imply. We study cases in which the latter two conjectures hold and disprove them for the general setting. (Shortened abstract due to arXiv restrictions.)

A drawing of a graph is x-monotone if every vertical line intersects each edge of the graph at most once. We present an O(n^5) time algorithm for deciding whether a simple drawing of the complete graph K_n is weakly isomorphic to an x-monotone drawing. We note that this algorithm can also decide whether a drawing of K_n is strongly isomorphic to an x-monotone drawing.

Given an outerplane drawing 𝒟 with n vertices, a 𝒟 -constrained maximum outerplane drawing is an outerplane drawing that contains 𝒟 and has the maximum number of edges among all such drawings. We show that (1) any such 𝒟 -constrained maximum outerplane drawing has at least n+1 edges, and (2) this bound is best possible. If 𝒟 is a plane spanning path, we present an O(n^3) -time algorithm to compute a 𝒟 -constrained maximum outerplane drawing that minimizes its weight, that is, the sum of the Euclidean length of the edges of its drawing. For the unweighted setting, our results imply the following: It can be decided in O(n^3) time whether a given plane spanning path admits a polygon such that every edge of the path is on the polygon or an internal edge. Further, we present for several minimum-weight structures, like minimum spanning trees, points sets such that they are not subdrawings of the minimum-weight outerplane Laman graph of that point set.

In a simple drawing of a graph, any two edges intersect in at most one point (either a common endpoint or a proper crossing). A simple drawing is generalized twisted if it fulfills certain rather specific constraints on how the edges are drawn. An abstract rotation system of a graph assigns to each vertex a cyclic order of its incident edges. A realizable rotation system is one that admits a simple drawing such that at each vertex, the edges emanate in that cyclic order, and a generalized twisted rotation system can be realized as a generalized twisted drawing. Generalized twisted drawings have initially been introduced to obtain improved bounds on the size of plane substructures in any simple drawing of K_n. They have since gained independent interest due to their surprising properties. However, the definition of generalized twisted drawings is very geometric and drawing-specific. In this paper, we develop characterizations of generalized twisted drawings that enable a purely combinatorial view on these drawings and lead to efficient recognition algorithms. Concretely, we show that for any n ≥ 7, an abstract rotation system of K_n is generalized twisted if and only if all subrotation systems induced by five vertices are generalized twisted. This implies a drawing-independent and concise characterization of generalized twistedness. Besides, the result yields a simple O(n^5)-time algorithm to decide whether an abstract rotation system is generalized twisted and sheds new light on the structural features of simple drawings. We further develop a characterization via the rotations of a pair of vertices in a drawing, which we then use to derive an O(n^2)-time algorithm to decide whether a realizable rotation system is generalized twisted.

A connected topological drawing of a graph divides the plane into a number of cells. The type of a cell $c$ is the cyclic sequence of crossings and vertices along the boundary walk of $c$. For example, all triangular cells with three incident crossings and no incident vertex share the same cell type. When a non-homotopic drawing of an $n$-vertex multigraph $G$ does not contain any such cells, Ackerman and Tardos [JCTA 2007] proved that $G$ has at most $8n-20$ edges, while Kaufmann, Klemz, Knorr, Reddy, Schröder, and Ueckerdt [GD 2024] showed that this bound is tight. In this paper, we initiate the in-depth study of non-homotopic drawings that do not contain one fixed cell type \mathfrak{c}, and investigate the edge density of the corresponding multigraphs, i.e., the maximum possible number of edges. We consider non-homotopic as well as simple drawings, multigraphs as well as simple graphs, and every possible type of cell. For every combination of drawing style, graph type, and cell type, we give upper and lower bounds on the corresponding edge density. With the exception of the cell type with four incident crossings and no incident vertex, we show for every cell type \mathfrak{c} that the edge density of $n$-vertex (multi)graphs with \mathfrak{c}-free drawings is either linear in $n$ or superlinear in $n$. In most cases, our bounds are tight up to an additive constant. We further consider cell types that are not incident to any crossing in more detail and find that all connected simple graphs but a short list of exceptions admit a simple drawing that does not contain any such cells. Additionally, we improve the current lower bound on the edge density of simple graphs that admit a non-homotopic quasiplanar drawing from $7n-28$ to $7.5n-28$.

A flip in a plane spanning tree T is the operation of removing one edge from T and adding another edge such that the resulting structure is again a plane spanning tree. For trees on a set of points in convex position we study two classic types of constrained flips: (1) Compatible flips are flips in which the removed and inserted edge do not cross each other. We relevantly improve the previous upper bound of 2n-O(√(n)) on the diameter of the compatible flip graph to 5n/3-O(1), by this matching the upper bound for unrestricted flips by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber [SODA 2025] up to an additive constant of 1. We further show that no shortest compatible flip sequence removes an edge that is already in its target position. Using this so-called happy edge property, we derive a fixed-parameter tractable algorithm to compute the shortest compatible flip sequence between two given trees. (2) Rotations are flips in which the removed and inserted edge share a common vertex. Besides showing that the happy edge property does not hold for rotations, we improve the previous upper bound of 2n-O(1) for the diameter of the rotation graph to 7n/4-O(1).

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.

We say that a (multi)graph $$ \user2{G} = (\user2{V},\user2{E}) $$ has geometric thickness t if there exists a straight-line drawing $$ \user2{\varphi }:\user2{V} \to \mathbb{R}^{{\mathbf{2}}} $$ and a t -coloring of its edges where no two edges sharing a point in their relative interior have the same color. The Geometric Thickness problem asks whether a given multigraph has geometric thickness at most t . This problem was shown to be NP-hard for $$ \user2{t} = \mathbf{2} $$ (Durocher et al. Comput Geom 56:1–18, 2016. https://doi.org/10.1016/j.comgeo.2016.03.003 ). In this paper, we settle the computational complexity of Geometric Thickness by showing that it is $$\exists \mathbb {R}$$ -complete already for thickness 30 . Moreover, our reduction shows that the problem is $$\exists \mathbb {R}$$ -complete for 4392 -planar graphs, where a graph is k -planar if it admits a topological drawing with at most k crossings per edge. In the course of our paper we answer previous questions on geometric thickness and on other related problems, in particular that simultaneous graph embeddings of 31 edge-disjoint graphs and pseudo-segment stretchability with chromatic number 30 are $$\exists \mathbb {R}$$ -complete.

For a finite set P of points in the plane in general position, a crossing family of size k in P is a collection of k line segments with endpoints in P that are pairwise crossing. It is a long-standing open problem to determine the largest size of a crossing family in any set of n points in the plane in general position. It is widely believed that this size should be linear in n. Motivated by results from the theory of partitioning complete geometric graphs, we study a variant of this problem for point sets P that do not contain a non-crossing family of size m, which is a collection of 4 disjoint subsets P_1, P_2, P_3, and P_4 of P, each containing m points of P, such that for every choice of 4 points p_i ∈ P_i, the set {p_1,p_2,p_3,p_4} is such that p_4 is in the interior of the triangle formed by p_1,p_2,p_3. We prove that, for every m ∈ℕ, each set P of n points in the plane in general position contains either a crossing family of size n/2^O(√(logm)) or a non-crossing family of size m, by this strengthening a recent breakthrough result by Pach, Rubin, and Tardos (2021). Our proof is constructive and we show that these families can be obtained in expected time O(nm^1+o(1)). We also prove that a crossing family of size Ω(n/m) or a non-crossing family of size m in P can be found in expected time O(n).

We study the geometric k-colored crossing number of complete graphs cr_k(K_n), which is the smallest number of monochromatic crossings in any k-edge colored straight-line drawing of K_n. We substantially improve asymptotic upper bounds on cr_k(K_n) for k=2,…, 10 by developing a procedure for general k that derives k-edge colored drawings of K_n for arbitrarily large n from initial drawings with a low number of monochromatic crossings. We obtain the latter by heuristic search, employing a MAX-k-CUT-formulation of a subproblem in the process.

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 one edge flip. The diameter diam(F(P)) of this graph is subject of intensive study. For points in general position, it is between 3n/2-5 and 2n-4, with no improvement for 25 years. For points in convex position, it lies between 3n/2 - 5 and ≈1.95n, where the lower bound was conjectured to be tight up to an additive constant and the upper bound is a recent breakthrough improvement over several bounds of the form 2n-o(n). In this work, we provide new upper and lower bounds on diam(F(P)), mainly focusing on points in convex position. We show 14n/9 - O(1) ≤ diam(F(P)) ≤ 5n/3 - 3, by this disproving the conjectured upper bound of 3n/2 for convex position, and relevantly improving both the long-standing lower bound for general position and the recent new upper bound for convex position. We complement these by showing that if one of T,T' has at most two boundary edges, then dist(T,T') ≤ 2d/2 < 3n/2, where d = |T-T'| is the number of edges in one tree that are not in the other. To prove both the upper and the lower bound, we introduce a new powerful tool. Specifically, we convert the flip distance problem for given T,T' to the problem of a largest acyclic subset in an associated conflict graph H(T,T'). In fact, this method is powerful enough to give an equivalent formulation of the diameter of F(P) for points P in convex position up to lower-order terms. As such, conflict graphs are likely the key to a complete resolution of this and possibly also other reconfiguration problems.

We study the \emph{geometric $k$-colored crossing number} of complete graphs $\overline{\overline{\text{cr}}}_k(K_n)$, which is the smallest number of monochromatic crossings in any $k$-edge colored straight-line drawing of $K_n$. We substantially improve asymptotic upper bounds on $\overline{\overline{\text{cr}}}_k(K_n)$ for $k=2,\ldots, 10$ by developing a procedure for general $k$ that derives $k$-edge colored drawings of $K_n$ for arbitrarily large $n$ from initial drawings with a low number of monochromatic crossings. We obtain the latter by heuristic search, employing a \textsc{MAX-$k$-CUT}-formulation of a subproblem in the process.

Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). A simple drawing is c-monotone if there is a point O such that each ray emanating from O crosses each edge of the drawing at most once. We introduce a special kind of c-monotone drawings that we call generalized twisted drawings. A c-monotone drawing is generalized twisted if there is a ray emanating from O that crosses all the edges of the drawing. Via this class of drawings, we show that every simple drawing of the complete graph with n vertices contains Ω (n^1/2) pairwise disjoint edges and a plane cycle (and hence path) of length Ω (log n /loglog n) . Both results improve over best previously published lower bounds. On the way we show several structural results and properties of generalized twisted and c-monotone drawings, some of which we believe to be of independent interest. For example, we show that a drawing D is c-monotone if there exists a point O such that no edge of D is crossed more than once by any ray that emanates from O and passes through a vertex of D .

We say that a (multi)graph G = (V,E) has geometric thickness t if there exists a straight-line drawing φ : V →ℝ^2 and a t-coloring of its edges where no two edges sharing a point in their relative interior have the same color. The Geometric Thickness problem asks whether a given multigraph has geometric thickness at most t. In this paper, we settle the computational complexity of Geometric Thickness by showing that it is ∃ℝ -complete already for thickness 57 . Moreover, our reduction shows that the problem is ∃ℝ -complete for 8280 -planar graphs, where a graph is k-planar if it admits a topological drawing with at most k crossings per edge. In this paper we answer previous questions on geometric thickness and on other related problems, in particular that simultaneous graph embeddings of 58 edge-disjoint graphs and pseudo-segment stretchability with chromatic number 57 are ∃ℝ -complete.

Visualizing a graph G in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masarik and Hlineny [GD 2023] recently asked for each edge of G to be drawn without crossings while allowing multiple different drawings of G. More formally, a collection D of drawings of G is uncrossed if, for each edge e of G, there is a drawing in D such that e is uncrossed. The uncrossed number unc(G) of G is then the minimum number of drawings in some uncrossed collection of G. No exact values of the uncrossed numbers have been determined yet, not even for simple graph classes. In this paper, we provide the exact values for uncrossed numbers of complete and complete bipartite graphs, partly confirming and partly refuting a conjecture posed by Hlineny and Masarik [GD 2023]. We also present a strong general lower bound on unc(G) in terms of the number of vertices and edges of G. Moreover, we prove NP-hardness of the related problem of determining the edge crossing number of a graph G, which is the smallest number of edges of G taken over all drawings of G that participate in a crossing. This problem was posed as open by Schaefer in his book [Crossing Numbers of Graphs 2018].

Simple drawings are drawings of graphs in which any two edges intersect at most once (either at a common endpoint or a proper crossing), and no edge intersects itself. We analyze several characteristics of simple drawings of complete multipartite graphs: which pairs of edges cross, in which order they cross, and the cyclic order around vertices and crossings, respectively. We consider all possible combinations of how two drawings can share some characteristics and determine which other characteristics they imply and which they do not imply. Our main results are that for simple drawings of complete multipartite graphs, the orders in which edges cross determine all other considered characteristics. Further, if all partition classes have at least three vertices, then the pairs of edges that cross determine the rotation system and the rotation around the crossings determine the extended rotation system. We also show that most other implications – including the ones that hold for complete graphs – do not hold for complete multipartite graphs. Using this analysis, we establish which types of isomorphisms are meaningful for simple drawings of complete multipartite graphs.

Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most 3. Motivated by this conjecture, we study the colorability of arrangement graphs for different classes of (pseudo-)circle arrangements. In this paper the conjecture is verified for △-saturated pseudocircle arrangements, i.e., for arrangements where one color class of the 2-coloring of faces consists of triangles only, as well as for further classes of (pseudo-)circle arrangements. These results are complemented by a construction which maps △-saturated arrangements with a pentagonal face to arrangements with 4-chromatic 4-regular arrangement graphs. This corona construction has similarities with the crowning construction introduced by Koester (1985). Based on exhaustive experiments with small arrangements we propose three strengthenings of the original conjecture. We further investigate fractional colorings. It is shown that the arrangement graph of every arrangement A of pairwise intersecting pseudocircles is “close” to being 3-colorable. More precisely, the fractional chromatic number χ f ( A ) of the arrangement graph is bounded from above by χ f ( A ) ≤ 3 + O ( 1 n ), where n is the number of pseudocircles of A. Furthermore, we construct an infinite family of 4-edge-critical 4-regular planar graphs which are fractionally 3-colorable. This disproves a conjecture of Gimbel, Kündgen, Li, and Thomassen (2019).