arXiv subjects
Publications and source records attributed to Alexander Pilz.
A drawing of a graph is {\em $x$-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 two distinct point sets $P$ and $Q$ in the plane, we say that $Q$ \emph{blocks} $P$ if no two points of $P$ are adjacent in any Delaunay triangulation of $P\cup Q$. Aichholzer et al. (2013) showed that any set $P$ of $n$ points in general position can be blocked by $\frac{3}{2}n$ points and that every set $P$ of $n$ points in convex position can be blocked by $\frac{5}{4}n$ points. Moreover, they conjectured that, if $P$ is in convex position, $n$ blocking points are sufficient and necessary. The necessity was recently shown by Biniaz (2021) who proved that every point set in general position requires $n$ blocking points. Here we investigate the variant, where blocking points can only lie outside of the convex hull of the given point set. We show that $\frac{5}{4}n-O(1)$ such \emph{exterior-blocking} points are sometimes necessary, even if the given point set is in convex position. As a consequence we obtain that, if the conjecture of Aichholzer et al. for the original setting was true, then minimal blocking sets of some point configurations $P$ would have to contain points inside of the convex hull of $P$.
Topological drawings are representations of graphs in the plane, where vertices are represented by points, and edges by simple curves connecting the points. A drawing is simple if two edges intersect at most in a single point, either at a common endpoint or at a proper crossing. In this paper we study properties of maximal plane subgraphs of simple drawings $D_n$ of the complete graph $K_n$ on $n$ vertices. Our main structural result is that maximal plane subgraphs are 2-connected and what we call essentially 3-edge-connected. Besides, any maximal plane subgraph contains at least $\lceil 3n/2 \rceil$ edges. We also address the problem of obtaining a plane subgraph of $D_n$ with the maximum number of edges, proving that this problem is NP-complete. However, given a plane spanning connected subgraph of $D_n$, a maximum plane augmentation of this subgraph can be found in $O(n^3)$ time. As a side result, we also show that the problem of finding a largest compatible plane straight-line graph of two labeled point sets is NP-complete.
A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of $n$ points there exists a compatible matching with $\lfloor \sqrt {2n}\rfloor$ edges. More generally, for any $\ell$ labeled point sets we construct compatible matchings of size $\Omega(n^{1/\ell})$. As a corresponding upper bound, we use probabilistic arguments to show that for any $\ell$ given sets of $n$ points there exists a labeling of each set such that the largest compatible matching has ${\mathcal{O}}(n^{2/({\ell}+1)})$ edges. Finally, we show that $\Theta(\log n)$ copies of any set of $n$ points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.
In order to have a compact visualization of the order type of a given point set S, we are interested in geometric graphs on S with few edges that unambiguously display the order type of S. We introduce the concept of exit edges, which prevent the order type from changing under continuous motion of vertices. That is, in the geometric graph on S whose edges are the exit edges, in order to change the order type of S, at least one vertex needs to move across an exit edge. Exit edges have a natural dual characterization, which allows us to efficiently compute them and to bound their number.
We study noncrossing geometric graphs and their disjoint compatible geometric matchings. Given a cycle (a polygon) P we want to draw a set of pairwise disjoint straight-line edges with endpoints on the vertices of P such that these new edges neither cross nor contain any edge of the polygon. We prove NP-completeness of deciding whether there is such a perfect matching. For any n-vertex polygon, with n > 3, we show that such a matching with less than n/7 edges is not maximal, that is, it can be extended by another compatible matching edge. We also construct polygons with maximal compatible matchings with n/7 edges, demonstrating the tightness of this bound. Tight bounds on the size of a minimal maximal compatible matching are also obtained for the families of d-regular geometric graphs for each d in {0,1,2}. Finally we consider a related problem. We prove that it is NP-complete to decide whether a noncrossing geometric graph G admits a set of compatible noncrossing edges such that G together with these edges has minimum degree five.
The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and planar straight-line graphs. For the latter, several variants have been studied (e.g., edge slides and edge rotations). In a transition graph on the set $\mathcal{T}(S)$ of noncrossing straight-line spanning trees on a finite point set $S$ in the plane, two spanning trees are connected by an edge if one can be transformed into the other by such an operation. We study bounds on the diameter of these graphs, and consider the various operations on point sets in both general position and convex position. In addition, we address variants of the problem where operations may be performed simultaneously or the edges are labeled. We prove new lower and upper bounds for the diameters of the corresponding transition graphs and pose open problems.
We consider arrangements of $n$ pseudo-lines in the Euclidean plane where each pseudo-line $\ell_i$ is represented by a bi-infinite connected $x$-monotone curve $f_i(x)$, $x \in \mathbb{R}$, s.t.\ for any two pseudo-lines $\ell_i$ and $\ell_j$ with $i < j$, the function $x \mapsto f_j(x) - f_i(x)$ is monotonically decreasing and surjective (i.e., the pseudo-lines approach each other until they cross, and then move away from each other). We show that such \emph{arrangements of approaching pseudo-lines}, under some aspects, behave similar to arrangements of lines, while for other aspects, they share the freedom of general pseudo-line arrangements. For the former, we prove: 1. There are arrangements of pseudo-lines that are not realizable with approaching pseudo-lines. 2. Every arrangement of approaching pseudo-lines has a dual generalized configuration of points with an underlying arrangement of approaching pseudo-lines. For the latter, we show: 1. There are $2^{Θ(n^2)}$ isomorphism classes of arrangements of approaching pseudo-lines (while there are only $2^{Θ(n \log n)}$ isomorphism classes of line arrangements). 2. It can be decided in polynomial time whether an allowable sequence is realizable by an arrangement of approaching pseudo-lines. Furthermore, arrangements of approaching pseudo-lines can be transformed into each other by flipping triangular cells, i.e., they have a connected flip graph, and every bichromatic arrangement of this type contains a bichromatic triangular cell.
We consider the following problem: Let $\mathcal{L}$ be an arrangement of $n$ lines in $\mathbb{R}^3$ colored red, green, and blue. Does there exist a vertical plane $P$ such that a line on $P$ simultaneously bisects all three classes of points in the cross-section $\mathcal{L} \cap P$? Recently, Schnider [SoCG 2019] used topological methods to prove that such a cross-section always exists. In this work, we give an alternative proof of this fact, using only methods from discrete geometry. With this combinatorial proof at hand, we devise an $O(n^2\log^2(n))$ time algorithm to find such a plane and the bisector of the induced cross-section. We do this by providing a general framework, from which we expect that it can be applied to solve similar problems on cross-sections and kinetic points.
A set $P = H \cup \{w\}$ of $n+1$ points in general position in the plane is called a wheel set if all points but $w$ are extreme. We show that for the purpose of counting crossing-free geometric graphs on such a set $P$, it suffices to know the frequency vector of $P$. While there are roughly $2^n$ distinct order types that correspond to wheel sets, the number of frequency vectors is only about $2^{n/2}$. We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, triangulations, and many more. Based on that, the corresponding numbers of graphs can be computed efficiently. In particular, we rediscover an already known formula for $w$-embracing triangles spanned by $H$. Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point $w$ in a set $H$, i.e., the number of $w$-embracing simplices. While our previous arguments in the plane do not generalize easily, we show how to use similar ideas in $\mathbb{R}^d$ for any fixed $d$. The result is an $O(n^{d-1})$ time algorithm for computing the simplicial depth of a point $w$ in a set $H$ of $n$ points, improving on the previously best bound of $O(n^d\log n)$. Based on our result about simplicial depth, we can compute the number of facets of the convex hull of $n=d+k$ points in general position in $\mathbb{R}^d$ in time $O(n^{\max\{ω,k-2\}})$ where $ω\approx 2.373$, even though the asymptotic number of facets may be as large as $n^k$.
We show that the graph transformation problem of turning a simple graph into an Eulerian one by a minimum number of single edge switches is NP-hard. Further, we show that any simple Eulerian graph can be transformed into any other such graph by a sequence of 2-switches (i.e., exchange of two edge pairs), such that every intermediate graph is also Eulerian. However, finding the shortest such sequence also turns out to be an NP-hard problem.
Assume you have a pizza consisting of four ingredients (e.g., bread, tomatoes, cheese and olives) that you want to share with your friend. You want to do this fairly, meaning that you and your friend should get the same amount of each ingredient. How many times do you need to cut the pizza so that this is possible? We will show that two straight cuts always suffice. More formally, we will show the following extension of the well-known Ham-sandwich theorem: Given four mass distributions in the plane, they can be simultaneously bisected with two lines. That is, there exist two oriented lines with the following property: let $R^+_1$ be the region of the plane that lies to the positive side of both lines and let $R^+_2$ be the region of the plane that lies to the negative side of both lines. Then $R^+=R^+_1\cup R^+_2$ contains exactly half of each mass distribution.
Given a set of points in the plane, we want to establish a connection network between these points that consists of several disjoint layers. Motivated by sensor networks, we want that each layer is spanning and plane, and that no edge is very long (when compared to the minimum length needed to obtain a spanning graph). We consider two different approaches: first we show an almost optimal centralized approach to extract two graphs. Then we show a constant factor approximation for a distributed model in which each point can compute its adjacencies using only local information. In both cases the obtained layers are plane
We use the concept of production matrices to show that there exist sets of $n$ points in the plane that admit $Ω(42.11^n)$ crossing-free geometric graphs. This improves the previously best known bound of $Ω(41.18^n)$ by Aichholzer et al. (2007).
The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set $P$ of $n$ points in $\mathbb{R}^d$, there is a point $c$, not necessarily from $P$, such that each halfspace containing $c$ contains at least $\frac{n}{d+1}$ points of $P$. Such a point $c$ is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set $P$. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median. We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set $Q$ of (few) points such that every halfspace that contains one point of $Q$ contains a large fraction of the points of $P$ and every halfspace that contains more of $Q$ contains an even larger fraction of $P$. This setting is comparable to the well-studied concepts of weak $\varepsilon$-nets and weak $\varepsilon$-approximations, where it is stronger than the former but weaker than the latter.
For both triangulations of point sets and simple polygons, it is known that determining the flip distance between two triangulations is an NP-hard problem. To gain more insight into flips of triangulations and to characterize "where edges go" when flipping from one triangulation to another, flips in edge-labeled triangulations have lately attracted considerable interest. In a recent breakthrough, Lubiw, Masárová, and Wagner (in Proc. 33rd Symp. of Computational Geometry, 2017) prove the so-called "Orbit Conjecture" for edge-labeled triangulations and ask for the complexity of the flip distance problem in the edge-labeled setting. By revisiting and modifying the hardness proofs for the unlabeled setting, we show in this note that the flip distance problem is APX-hard for edge-labeled triangulations of point sets and NP-hard for triangulations of simple polygons. The main technical challenge is to show that this remains true even if the source and target triangulation are the same when disregarding the labeling.
Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-complete. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.
Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Nešetřil and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey. As our main result, we introduce a new family of $(k,2)$-Ramsey point sets, extending a result of Nešetřil and Valtr. We then use this new result to show that for every $k$ there is a point set $P$ such that no function $Γ$ that maps ordered pairs of distinct points from $P$ to a set of size $k$ can satisfy the following "local consistency" property: if $Γ$ attains the same values on two ordered triples of points from $P$, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.