We study the problem of reconfiguring a crossing-free embedding of a graph on a surface, with edges represented as curves, into another crossing-free embedding of the same graph on the same surface with the same fixed vertex positions. In this process, we reroute one edge at a time while maintaining crossing-free intermediate embeddings. This problem was introduced by Ito et al. [TALG 2025], who showed that even if the graph is a matching of two edges, reconfiguration is not always possible in the plane, but is always possible on the torus. For matchings of two or more edges, they gave a necessary and sufficient condition for reconfigurable embeddings in the plane, but not on the torus. Our main result is that for matchings, trees and forests, reconfiguration is always possible on the torus, and consequently, on any orientable surface of genus at least one. In addition, we provide sufficient conditions for reconfiguration on orientable surfaces of genus at least one and in the projective plane. For more general graphs, we show that reconfiguration is not always possible.

The visibility graph of a set S ⊆ ℝ² is the graph whose vertices are the points of S, with two points x,y connected by an edge if and only if they see each other in S, that is, if the segment xy is contained in S. The edge density of this graph is known as the Beer index of S. Previously, it has been shown that a simply connected set S ⊆ ℝ² of unit Lebesgue measure with Beer index β > 0 contains a convex subset of measure Ω(β); in particular, for visibility graphs of simply connected sets, a positive edge density β > 0 implies the existence of a clique containing an Ω(β)-fraction of all vertices. The simple-connectivity assumption cannot be omitted, as there are non-simply-connected sets with Beer index 1 and no convex subset of positive measure. Nevertheless, in this paper, we extend the above result to non-simply-connected sets, by showing that a visibility graph with large edge density contains a triangle with large convex hull. More precisely, we show that a set S ⊆ ℝ² of unit Lebesgue measure with Beer index β > 0 contains three pairwise visible points whose convex hull has measure Ω(β⁹). If in addition S is an open domain with K holes, then S contains three pairwise visible points with convex hull of measure Ω(β/K) as well as a convex subset of measure Ω(β/K²).

Abstract. An obstacle representation of a graph [Formula: see text] consists of a set of pairwise disjoint simply connected closed regions and a one-to-one mapping of the vertices of [Formula: see text] to points such that two vertices are adjacent in [Formula: see text] if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each [Formula: see text]-vertex graph is [Formula: see text] [M. Balko, J. Cibulka, and P. Valtr, Discrete Comput. Geom., 59 (2018), pp. 143–164] and that there are [Formula: see text]-vertex graphs whose obstacle number is [Formula: see text] [V. Dujmović and P. Morin, Electron. J. Combin., 22 (2015), 3.1]. We improve this lower bound to [Formula: see text] for simple polygons and to [Formula: see text] for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of [Formula: see text]-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some [Formula: see text]-vertex graph is given as part of the input, then for some drawings [Formula: see text] obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph [Formula: see text] is fixed-parameter tractable in the vertex cover number of [Formula: see text]. Second, we show that, given a graph [Formula: see text] and a simple polygon [Formula: see text], it is NP-hard to decide whether [Formula: see text] admits an obstacle representation using [Formula: see text] as the only obstacle.

We consider point sets in the real projective plane ℝ𝒫² and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdős-Szekeres-type problems. We provide asymptotically tight bounds for a variant of the Erdős-Szekeres theorem about point sets in convex position in ℝ𝒫², which was initiated by Harborth and Möller in 1994. The notion of convex position in ℝ𝒫² agrees with the definition of convex sets introduced by Steinitz in 1913. For k ≥ 3, an (affine) k-hole in a finite set S ⊆ ℝ² is a set of k points from S in convex position with no point of S in the interior of their convex hull. After introducing a new notion of k-holes for points sets from ℝ𝒫², called projective k-holes, we find arbitrarily large finite sets of points from ℝ𝒫² with no projective 8-holes, providing an analogue of a classical result by Horton from 1983. We also prove that they contain only quadratically many projective k-holes for k ≤ 7. On the other hand, we show that the number of k-holes can be substantially larger in ℝ𝒫² than in ℝ² by constructing, for every k ∈ {3,… ,6}, sets of n points from ℝ² ⊂ ℝ𝒫² with Ω(n^{3-3/5k}) projective k-holes and only O(n²) affine k-holes. Last but not least, we prove several other results, for example about projective holes in random point sets in ℝ𝒫² and about some algorithmic aspects. The study of extremal problems about point sets in ℝ𝒫² opens a new area of research, which we support by posing several open problems.

For a finite set A ⊂ ℝ², a map φ: A → ℝ² is orientation preserving if for every non-collinear triple u,v,w ∈ A the orientation of the triangle u,v,w is the same as that of the triangle φ(u),φ(v),φ(w). We prove that for every n ∈ ℕ and for every ε > 0 there is N = N(n,ε) ∈ ℕ such that the following holds. Assume that φ:G(N) → ℝ² is an orientation preserving map where G(N) is the grid {(i,j) ∈ ℤ²: -N ≤ i,j ≤ N}. Then there is an affine transformation ψ :ℝ² → ℝ² and a ∈ ℤ² such that a+G(n) ⊂ G(N) and ‖ψ∘φ (z)-z‖ < ε for every z ∈ a+G(n). This result was previously proved in a completely different way by Nešetřil and Valtr, without obtaining any bound on N. Our proof gives N(n,ε) = O(n⁴ε^{-2}).

Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from $P$ so that (i) all points are pairwise distinct; (ii) any two consecutive points $p_i$, $p_{i+1}$ have different colors; and (iii) any two segments $p_i p_{i+1}$ and $p_j p_{j+1}$ have disjoint relative interiors, for $i \neq j$. We show that there is an absolute constant $\varepsilon > 0$, independent of $n$ and of the coloring, such that $P$ always admits a non-crossing alternating path of length at least $(1 + \varepsilon)n$. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least $(1 + \varepsilon)n$ points of $P$. This is a properly colored matching whose segments are pairwise disjoint and intersected by common line. For both versions, this is the first improvement of the easily obtained lower bound of $n$ by an additive term linear in $n$. The best known published upper bounds are asymptotically of order $4n/3+o(n)$.

Let d and k be integers with $$1 \le k \le d-1$$ . Let $$\Lambda $$ be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in $$\Lambda \cap K$$ . In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional $$n \times \cdots \times n$$ grid is at least $$\Omega \bigl (n^{d(d-k)/(d-1)-\varepsilon }\bigr )$$ and at most $$O\bigl (n^{d(d-k)/(d-1)}\bigr )$$ , where $$\varepsilon >0$$ is an arbitrarily small constant. This nearly settles a problem mentioned in the book by Brass et al. (Research problems in discrete geometry, Springer, New York, 2005). We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover $$\Lambda \cap K$$ . We use these new results to improve the best known lower bound for the maximum number of point–hyperplane incidences by Brass and Knauer (Comput Geom 25(1–2):13–20, 2003). For $$d \ge 3$$ and $$\varepsilon \in (0,1)$$ , we show that there is an integer $$r=r(d,\varepsilon )$$ such that for all positive integers n, m the following statement is true. There is a set of n points in $$\mathbb {R}^d$$ and an arrangement of m hyperplanes in $$\mathbb {R}^d$$ with no $$K_{r,r}$$ in their incidence graph and with at least $$\Omega \bigl ((mn)^{1-(2d+3)/((d+2)(d+3)) - \varepsilon }\bigr )$$ incidences if d is odd and $$\Omega \bigl ((mn)^{1-(2d^2+d-2)/((d+2)(d^2+2d-2)) -\varepsilon }\bigr )$$ incidences if d is even.

A permutation pi is a merge of a permutation sigma and a permutation tau, if we can color the elements of pi red and blue so that the red elements have the same relative order as sigma and the blue ones as tau. We consider, for fixed hereditary permutation classes C and D, the complexity of determining whether a given permutation pi is a merge of an element of C with an element of D. We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of nondeterministic logspace streaming recognizability of permutations, which we introduce, and a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich. As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations, the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length. On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the pattern 2413.

Let S be a subset of $$\mathbb {R}^d$$ with finite positive Lebesgue measure. The Beer index of convexity $${\text {b}}(S)$$ of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio $${\text {c}}(S)$$ of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate the relationship between these two natural measures of convexity. We show that every set $$S\subseteq \mathbb {R}^2$$ with simply connected components satisfies $${\text {b}}(S)\leqslant \alpha {\text {c}}(S)$$ for an absolute constant $$\alpha $$ , provided $${\text {b}}(S)$$ is defined. This implies an affirmative answer to the conjecture of Cabello et al. that this estimate holds for simple polygons. We also consider higher-order generalizations of $${\text {b}}(S)$$ . For $$1\leqslant k\leqslant d$$ , the k-index of convexity $${\text {b}}_k(S)$$ of a set $$S\subseteq \mathbb {R}^d$$ is the probability that the convex hull of a $$(k+1)$$ -tuple of points chosen uniformly independently at random from S is contained in S. We show that for every $$d\geqslant 2$$ there is a constant $$\beta (d)>0$$ such that every set $$S\subseteq \mathbb {R}^d$$ satisfies $${\text {b}}_d(S)\leqslant \beta {\text {c}}(S)$$ , provided $${\text {b}}_d(S)$$ exists. We provide an almost matching lower bound by showing that there is a constant $$\gamma (d)>0$$ such that for every $$\varepsilon \in (0,1)$$ there is a set $$S\subseteq \mathbb {R}^d$$ of Lebesgue measure 1 satisfying $${\text {c}}(S)\leqslant \varepsilon $$ and $${\text {b}}_d(S)\geqslant \gamma \frac{\varepsilon }{\log _2{1/\varepsilon }}\geqslant \gamma \frac{{\text {c}}(S)}{\log _2{1/{\text {c}}(S)}}$$ .

We consider the following geometric optimization problem: find a convex polygon of maximum area contained in a given simple polygon $P$ with $n$ vertices. We give a randomized near-linear-time $(1-\varepsilon)$-approximation algorithm for this problem: in $O(n( \log^2 n + (1/\varepsilon^3) \log n + 1/\varepsilon^4))$ time we find a convex polygon contained in $P$ that, with probability at least $2/3$, has area at least $(1-\varepsilon)$ times the area of an optimal solution. We also obtain similar results for the variant of computing a convex polygon inside $P$ with maximum perimeter. To achieve these results we provide new results in geometric probability. The first result is a bound relating the area of the largest convex body inside $P$ to the probability that two points chosen uniformly at random inside $P$ are mutually visible. The second result is a bound on the expected value of the difference between the perimeter of any planar convex body $K$ and the perimeter of the convex hull of a uniform random sample inside $K$.

Let S be a subset of R^d with finite positive Lebesgue measure. The Beer index of convexity b(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio c(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate a relationship between these two natural measures of convexity of S. We show that every subset S of the plane with simply connected components satisfies b(S) <= alpha c(S) for an absolute constant alpha, provided b(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. asserting that this estimate holds for simple polygons. We also consider higher-order generalizations of b(S). For 1 <= k <= d, the k-index of convexity b_k(S) of a subset S of R^d is the probability that the convex hull of a (k+1)-tuple of points chosen uniformly independently at random from S is contained in S. We show that for every d >= 2 there is a constant beta(d) > 0 such that every subset S of R^d satisfies b_d(S) <= beta c(S), provided b_d(S) exists. We provide an almost matching lower bound by showing that there is a constant gamma(d) > 0 such that for every epsilon from (0,1] there is a subset S of R^d of Lebesgue measure one satisfying c(S) <= epsilon and b_d(S) >= (gamma epsilon)/log_2(1/epsilon) >= (gamma c(S))/log_2(1/c(S)).

An empty pentagon in a point set $P$ in the plane is a set of five points in $P$ in strictly convex position with no other point of $P$ in their convex hull. We prove that every finite set of at least $328\ell^2$ points in the plane contains an empty pentagon or $\ell$ collinear points. This is optimal up to a constant factor since the $(\ell -1)\times(\ell-1)$ square lattice contains no empty pentagon and no $\ell$ collinear points. The previous best known bound was doubly exponential.

We consider the following geometric optimization problem: find a convex polygon of maximum area contained in a given simple polygon P with n vertices. We give a randomized near-linear-time (1 − ϵ)-approximation algorithm for this problem: in O((n/ϵ6) log2 n log(1/δ)) time we find a convex polygon contained in P that, with probability at least 1 − δ, has area at least (1 − ϵ) times the area of an optimal solution.

Motivated by recent papers of Eppstein, Abel et al. and Barat et al., and by a question of Wood, we investigate properties of planar point sets with no 5-hole (no empty convex pentagon). We answer a question of Wood by showing that the visibility graph of a finite point set with no 5-hole may contain a clique of arbitrary size. This is in contrast with the previous examples of sets with no 5-hole, including the example of the (finite) square lattice. In our construction we use several equivalent local characterizations of (locally) finite planar point sets with no 5-hole which may be of independent interest. Our construction relies on a construction scheme which allows to derive new, non-trivial examples of sets with no 5-hole.

Let P be a set of n ≥ 4 points in the plane that is in general position and such that n is even. We investigate the problem whether there is a cubic plane straight-line graph on P. No polynomial-time algorithm is known for this problem. Based on a reduction to the existence of certain diagonals of the boundary cycle of the convex hull of P, we give the first polynomial-time algorithm; the algorithm is constructive and runs in time O(n3). We also show which graph structure can be expected when there is a cubic plane graph on P; e.g., if P admits a 2-connected cubic plane graph, we show that P admits also a 2-connected cubic plane graph that contains the boundary cycle of P. The algorithm extends to checking P on admitting a 2-connected cubic plane graph.

Giving a (partial) solution to a problem of Fekete [Geometry and the Traveling Salesman Problem, Ph.D. thesis, University of Waterloo, Waterloo, ON, Canada, 1992] and Fekete and Woeginger [Comput. Geom., 8 (1997), pp. 195–218], we show that given a finite set X of points in the plane, it is possible to find a polygonal path with $|X|-1$ segments and with vertex set X so that every angle on the polygonal path is at least $\pi/9$. According to a conjecture of Fekete and Woeginger, $\pi/9$ can be replaced by $\pi/6$. Previously, the result has not been known with any positive constant. We show further that the same result holds, with an angle smaller than $\pi/9$, in higher dimensions.

Given a finite set of points S in Rd, consider visiting thepoints in S with a polygonal path that makes a minimum number ofturns, or equivalently, has the the minimum number of segments(links). We call this minimization problem the minimum linkspanning path problem. This natural problem has appeared severaltimes in the literature under different variants. The simplest oneis where the allowed paths are axis-aligned. Let L(S) be theminimum number of links of an axis-aligned path for S denote by Gdn the d-dimensional grid of size n. Kranakis, Krizanc andMeertens (Ars Combinatoria, vol. 38, pp. 177--192, 1994)showed that in 2-dimensions L(G2n)=2n-1 and in three dimensions 4/3 n2-O(n)< L(G3n) < 3/2 n2+O(n). Kranakiset al. conjectured that, for all d ≥ 3, L(Gdn)= d/d-1 nd-1 ± O(nd-2). We prove theconjecture for d=3 by showing that L(G3n) ≥ 3/2 n2 -O(n). For d=4, we prove that 4/3 n3 -O(n2) ≤ L(G4n) ≤ 4/3 n3 +O(n5/2).For general d, we give new estimates on L(Gdn), that bring usvery close to the conjectured value. The new lower bound of (1+ 1/d)nd-1-O(nd-2) improves previous result byCollins and Moret (Information Processing Letters, vol. 68,pp. 317--319, 1998), while the new upper bound of (1+ 1/d-1)nd-1+O(nd-3/2) differs from the conjecturedvalue only in the lower order terms. For arbitrary point sets, we give an exact bound on the minimumnumber of links needed in an axis-aligned path traversing any planar n-point set. We obtain similar tight estimates (within 1) in anynumber of dimensions d. For the general problem of traversing anarbitrary set of points in Rd with an axis-aligned spanning pathhaving a minimum number of links, we present a constant ratio(depending on the dimension d) approximation algorithm.

Motivated by $L(p,q)$-labelings of graphs, we introduce a notion of $\lambda$-graphs: a $\lambda$-graph G is a graph with two types of edges: 1-edges and x-edges. For a parameter $x\in[0,1]$, a proper labeling of G is a labeling of vertices of G by nonnegative reals such that the labels of the endvertices of a 1-edge differ by at least 1 and the labels of the endvertices of an x-edge differ by at least x; $\lambda_G(x)$ is the smallest real such that G has a proper labeling by labels from the interval $[0,\lambda_G(x)]$. We study properties of the function $\lambda_G(x)$ for finite and infinite $\lambda$-graphs and establish the following results: if the function $\lambda_G(x)$ is well defined, then it is a piecewise linear function of x with finitely many linear parts. Surprisingly, the set $\Lambda(\alpha,\beta)$ of all functions $\lambda_G$ with $\lambda_G(0)=\alpha$ and $\lambda_G(1)=\beta$ is finite for any $\alpha\le\beta$. We also prove a tight upper bound on the number of segments for finite $\lambda$-graphs G with convex functions $\lambda_G(x)$.

In this paper, we study the Vapnik-Chervonenkis (VC)-dimension of set systems arising in 2D polygonal and 3D polyhedral configurations where a subset consists of all points visible from one camera. In the past, it has been shown that the VC-dimension of planar visibility systems is bounded by 23 if the cameras are allowed to be anywhere inside a polygon without holes. Here, we consider the case of exterior visibility, where the cameras lie on a constrained area outside the polygon and have to observe the entire boundary. We present results for the cases of cameras lying on a circle containing a polygon (VC-dimension= 2) or lying outside the convex hull of a polygon (VC-dimension= 5). The main result of this paper concerns the 3D case: We prove that the VC-dimension is unbounded if the cameras lie on a sphere containing the polyhedron, hence the term exterior visibility.

We analyze a randomized pivoting process involving one line and n points in the plane. The process models the behavior of the Random-Edge simplex algorithm on simple polytopes with n facets in dimension n-2. We obtain a tight O(\log^2 n) bound for the expected number of pivot steps. This is the first nontrivial bound for Random-Edge which goes beyond bounds for specific polytopes. The process itself can be interpreted as a simple algorithm for certain 2-variable linear programming problems, and we prove a tight t(n) bound for its expected runtime.The combinatorial structure behind the process is a directed graph over pairs of points, with arc orientations induced by the pivot steps. We characterize the class of graphs arising from one line and n points, up to oriented matroid realizability.

A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points. Improving a result of Pach and Töröcsik, we show that a geometric graph on n vertices with no k + 1 pairwise disjoint edges has at most k³(n + 1) edges. On the other hand, we construct geometric graphs with n vertices and approximately 3/2 (k - 1)n edges, containing no k + 1 pairwise disjoint edges. We also improve both the lower and upper bounds of Goddard, Katchalski and Kleitman on the maximum number of edges in a geometric graph with no four pairwise disjoint edges.

We show that for any 2-coloring of the ~) segments determined by n points in the plane, one of the color classes contains non-crossing cycles of lengths 3,4,.... [ ~j.This result is tight up to a multiplicative constant.Under the same assumptions, we also prove that there is a non-crossing path of length Q(n2f3), all of whose edges are of the same color.In the special case when the n points are in convex position, we find longer monochromatic non-crossing paths, of length [ ~1.This bound cannot be improved.All of these cycles and paths can be found by O(n2) time algorithms.We also discuss some related problems and generalizations.In particular, we give sharp estimates for the largest number of disjoint monochromatic triangles that can always be selected from our segments.

A halving hyperplane of a set S of n points in Rd contains d affinely independent points of S so that equally many of the points off the hyperplane lie in each of the two half-spaces. We prove bounds on the number of halving hyperplanes under the condition that the ratio of largest over smallest distance between any two points is at most δn1/d, δ some constant. Such a set S is called dense.