arXiv subjects

Birgit Vogtenhuber

Publications and source records attributed to Birgit Vogtenhuber.

At least 37 records · Page 2Linked to original sources

arXiv2023

Bichromatic Perfect Matchings with Crossings

We consider bichromatic point sets with $n$ red and $n$ blue points and study straight-line bichromatic perfect matchings on them. We show that every such point set in convex position admits a matching with at least $\frac{3n^2}{8}-\frac{n}{2}+c$ crossings, for some $ -\frac{1}{2} \leq c \leq \frac{1}{8}$. This bound is tight since for any $k> \frac{3n^2}{8} -\frac{n}{2}+\frac{1}{8}$ there exist bichromatic point sets that do not admit any perfect matching with $k$ crossings.

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arXiv2023

Different Types of Isomorphisms of Drawings of Complete Multipartite Graphs

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.

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arXiv2023

Adjacency Graphs of Polyhedral Surfaces

We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in $\mathbb{R}^3$. We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains $K_5$, $K_{5,81}$, or any nonplanar $3$-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, $K_{4,4}$, and $K_{3,5}$ can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (1983), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable $n$-vertex graphs is in $Ω(n \log n)$. From the non-realizability of $K_{5,81}$, we obtain that any realizable $n$-vertex graph has $O(n^{9/5})$ edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

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arXiv2023

Drawings of Complete Multipartite Graphs Up to Triangle Flips

For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph $K_n$ with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on $n$ vertices is bounded by $O(n^{16})$. The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the sense that having the same ERS does not remain sufficient when removing or adding very few edges.

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arXiv2022

Blocking Delaunay Triangulations from the Exterior

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 $P$.

cs.CG↗

arXiv2022

Graphs with large total angular resolution

The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing. It combines two properties contributing to the readability of a drawing: the angular resolution, which is the minimum angle between incident edges, and the crossing resolution, which is the minimum angle between crossing edges. We consider the total angular resolution of a graph, which is the maximum total angular resolution of a straight-line drawing of this graph. We prove that, up to a finite number of well specified exceptions of constant size, the number of edges of a graph with $n$ vertices and a total angular resolution greater than $60^{\circ}$ is bounded by $2n-6$. This bound is tight. In addition, we show that deciding whether a graph has total angular resolution at least $60^{\circ}$ is NP-hard.

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arXiv2022

No Selection Lemma for Empty Triangles

Let $S$ be a set of $n$ points in general position in the plane. The Second Selection Lemma states that for any family of $Θ(n^3)$ triangles spanned by $S$, there exists a point of the plane that lies in a constant fraction of them. For families of $Θ(n^{3-α})$ triangles, with $0\le α\le 1$, there might not be a point in more than $Θ(n^{3-2α})$ of those triangles. An empty triangle of $S$ is a triangle spanned by $S$ not containing any point of $S$ in its interior. Bárány conjectured that there exist an edge spanned by $S$ that is incident to a super constant number of empty triangles of $S$. The number of empty triangles of $S$ might be $O(n^2)$; in such a case, on average, every edge spanned by $S$ is incident to a constant number of empty triangles. The conjecture of Bárány suggests that for the class of empty triangles the above upper bound might not hold. In this paper we show that, somewhat surprisingly, the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer $n$ and real number $0\leq α\leq 1$ there exists a point set of size $n$ with $Θ(n^{3-α})$ empty triangles such that any point of the plane is only in $O(n^{3-2α})$ empty triangles.

cs.CG↗

arXiv2022

Flipping Plane Spanning Paths

Let $S$ be a planar point set in general position, and let $\mathcal{P}(S)$ be the set of all plane straight-line paths with vertex set $S$. A flip on a path $P \in \mathcal{P}(S)$ is the operation of replacing an edge $e$ of $P$ with another edge $f$ on $S$ to obtain a new valid path from $\mathcal{P}(S)$. It is a long-standing open question whether for every given point set $S$, every path from $\mathcal{P}(S)$ can be transformed into any other path from $\mathcal{P}(S)$ by a sequence of flips. To achieve a better understanding of this question, we show that it is sufficient to prove the statement for plane spanning paths whose first edge is fixed. Furthermore, we provide positive answers for special classes of point sets, namely, for wheel sets and generalized double circles (which include, e.g., double chains and double circles).

cs.CG↗

arXiv2022

On Compatible Matchings

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 $Ω(n^{1/\ell})$. 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 $Θ(\log n)$ 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.

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arXiv2022

Shooting Stars in Simple Drawings of $K_{m,n}$

Simple drawings are drawings of graphs in which two edges have at most one common point (either a common endpoint, or a proper crossing). It has been an open question whether every simple drawing of a complete bipartite graph $K_{m,n}$ contains a plane spanning tree as a subdrawing. We answer this question to the positive by showing that for every simple drawing of $K_{m,n}$ and for every vertex $v$ in that drawing, the drawing contains a shooting star rooted at $v$, that is, a plane spanning tree containing all edges incident to $v$.

cs.CG↗

arXiv2022

Compatible Spanning Trees in Simple Drawings of $K_n$

For a simple drawing $D$ of the complete graph $K_n$, two (plane) subdrawings are compatible if their union is plane. Let $\mathcal{T}_D$ be the set of all plane spanning trees on $D$ and $\mathcal{F}(\mathcal{T}_D)$ be the compatibility graph that has a vertex for each element in $\mathcal{T}_D$ and two vertices are adjacent if and only if the corresponding trees are compatible. We show, on the one hand, that $\mathcal{F}(\mathcal{T}_D)$ is connected if $D$ is a cylindrical, monotone, or strongly c-monotone drawing. On the other hand, we show that the subgraph of $\mathcal{F}(\mathcal{T}_D)$ induced by stars, double stars, and twin stars is also connected. In all cases the diameter of the corresponding compatibility graph is at most linear in $n$.

math.CO↗

arXiv2022

Empty Triangles in Generalized Twisted Drawings of $K_n$

Simple drawings are drawings of graphs in the plane or on the sphere such that vertices are distinct points, edges are Jordan arcs connecting their endpoints, and edges intersect at most once (either in a proper crossing or in a shared endpoint). Simple drawings are generalized twisted if there is a point $O$ such that every ray emanating from $O$ crosses every edge of the drawing at most once and there is a ray emanating from $O$ which crosses every edge exactly once. We show that all generalized twisted drawings of $K_n$ contain exactly $2n-4$ empty triangles, by this making a substantial step towards proving the conjecture that this is the case for every simple drawing of $K_n$.

cs.CG↗

arXiv2022

On Crossing-Families in Planar Point Sets

A $k$-crossing family in a point set $S$ in general position is a set of $k$ segments spanned by points of $S$ such that all $k$ segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4. (2) There are sets of $n$ points which do not contain a crossing family of size larger than $8\lceil \frac{n}{41} \rceil$. Both results improve the previously best known bounds.

cs.CG↗

arXiv2022

Coloring circle arrangements: New $4$-chromatic planar 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 arrangements of (pseudo-)circles. In this paper the conjecture is verified for $\triangle$-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 $\triangle$-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 also investigate fractional colorings. It is shown that the arrangement graph of every arrangement $\mathcal{A}$ of pairwise intersecting pseudocircles is "close" to being $3$-colorable. More precisely, the fractional chromatic number $χ_f(\mathcal{A})$ of the arrangement graph is bounded from above by $χ_f(\mathcal{A}) \le 3+O(\frac{1}{n})$, where $n$ is the number of pseudocircles of $\mathcal{A}$. 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).

math.CO↗

arXiv2022

Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

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). We introduce a special kind of simple drawings that we call generalized twisted drawings. A simple drawing is generalized twisted if there is a point $O$ such that every ray emanating from $O$ crosses every edge of the drawing at most once and there is a ray emanating from $O$ which crosses every edge exactly once. Via this new class of simple drawings, we show that every simple drawing of the complete graph with $n$ vertices contains $Ω(n^{\frac{1}{2}})$ pairwise disjoint edges and a plane path of length $Ω(\frac{\log n }{\log \log n})$. Both results improve over previously known best lower bounds. On the way we show several structural results about and properties of generalized twisted drawings. We further present different characterizations of generalized twisted drawings, which might be of independent interest.

cs.CG↗

arXiv2022

Inserting one edge into a simple drawing is hard

A {\em simple drawing} $D(G)$ of a graph $G$ is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge $e$ in the complement of $G$ can be {\em inserted} into $D(G)$ if there exists a simple drawing of $G+e$ extending $D(G)$. As a result of Levi's Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of $G$ can be inserted. In contrast, we show that it is NP -complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles $\mathcal{A}$ and a pseudosegment $σ$, it can be decided in polynomial time whether there exists a pseudocircle $Φ_σ$ extending $σ$ for which $\mathcal{A}\cup\{Φ_σ\}$ is again an arrangement of pseudocircles.

cs.CG↗

arXiv2021

Edge Partitions of Complete Geometric Graphs (Part 2)

Recently, the second and third author showed that complete geometric graphs on $2n$ vertices in general cannot be partitioned into $n$ plane spanning trees. Building up on this work, in this paper, we initiate the study of partitioning into beyond planar subgraphs, namely into $k$-planar and $k$-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

math.CO↗

arXiv2021

Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, \[ \frac{40\lfloor (k-1)/2 \rfloor -8}{19} %Birgit: \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11. \] Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time.

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