arXiv subjects

Stefan Felsner

Publications and source records attributed to Stefan Felsner.

At least 37 records · Page 2Linked to original sources

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).

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arXiv2021

Approximating the Bundled Crossing Number

Bundling crossings is a strategy which can enhance the readability of drawings. In this paper we consider good drawings, i.e., we require that any two edges have at most one common point which can be a common vertex or a crossing. Our main result is that there is a polynomial time algorithm to compute an 8-approximation of the bundled crossing number of a good drawing (up to adding a term depending on the facial structure of the drawing). In the special case of circular drawings the approximation factor is 8 (no extra term), this improves upon the 10-approximation of Fink et al. (Bundled crossings in embedded graphs, Proc. Latin'16). Our approach also works with the same approximation factor for families of pseudosegments, i.e., curves intersecting at most once. We also show how to compute a 9/2-approximation when the intersection graph of the pseudosegments is bipartite.

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arXiv2021

On the Queue-Number of Partial Orders

The queue-number of a poset is the queue-number of its cover graph viewed as a directed acyclic graph, i.e., when the vertex order must be a linear extension of the poset. Heath and Pemmaraju conjectured that every poset of width $w$ has queue-number at most $w$. Recently, Alam et al. constructed posets of width $w$ with queue-number $w+1$. Our contribution is a construction of posets with width $w$ with queue-number $Ω(w^2)$. This asymptotically matches the known upper bound.

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arXiv2021

Linear Layouts of Complete Graphs

A page (queue) with respect to a vertex ordering of a graph is a set of edges such that no two edges cross (nest), i.e., have their endpoints ordered in an ABAB-pattern (ABBA-pattern). A union page (union queue) is a vertex-disjoint union of pages (queues). The union page number (union queue number) of a graph is the smallest $ k $ such that there is a vertex ordering and a partition of the edges into $ k $ union pages (union queues). The local page number (local queue number) is the smallest $ k $ for which there is a vertex ordering and a partition of the edges into pages (queues) such that each vertex has incident edges in at most $ k $ pages (queues). We present upper and lower bounds on these four parameters for the complete graph $ K_n $ on $ n $ vertices. In three cases we obtain the exact result up to an additive constant. In particular, the local page number of $ K_n $ is $ n/3 \pm O(1) $, while its local and union queue number is $ (1-1/\sqrt{2})n \pm O(1) $. The union page number of $ K_n $ is between $ n/3 - O(1) $ and $ 4n/9 + O(1) $.

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arXiv2021

Improved bounds for centered colorings

A vertex coloring $ϕ$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $ϕ$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$-centered coloring using at most $f(p)$ colors. In this paper, we give upper bounds for the maximum number of colors needed in a $p$-centered coloring of graphs from several widely studied graph classes. We show that: (1) planar graphs admit $p$-centered colorings with $\mathcal{O}(p^3\log p)$ colors where the previous bound was $\mathcal{O}(p^{19})$; (2) bounded degree graphs admit $p$-centered colorings with $\mathcal{O}(p)$ colors while it was conjectured that they may require exponential number of colors in $p$; (3) graphs avoiding a fixed graph as a topological minor admit $p$-centered colorings with a polynomial in $p$ number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth $t$ that require $\binom{p+t}{t}$ colors in any $p$-centered coloring and this bound matches the upper bound; (5) there are planar graphs that require $Ω(p^2\log p)$ colors in any $p$-centered coloring.

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arXiv2021

Reconfiguring Independent Sets on Interval Graphs

We study reconfiguration of independent sets in interval graphs under the token sliding rule. We show that if two independent sets of size $k$ are reconfigurable in an $n$-vertex interval graph, then there is a reconfiguration sequence of length $\mathcal{O}(k\cdot n^2)$. We also provide a construction in which the shortest reconfiguration sequence is of length $Ω(k^2\cdot n)$. As a counterpart to these results, we also establish that $\textsf{Independent Set Reconfiguration}$ is PSPACE-hard on incomparability graphs, of which interval graphs are a special case.

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arXiv2020

On the Maximum Number of Crossings in Star-Simple Drawings of $K_n$ with No Empty Lens

A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced by two edges that is bounded by a 2-cycle), two independent edges may cross arbitrarily many times in a star-simple drawing. We consider star-simple drawings of $K_n$ with no empty lens. In this setting we prove an upper bound of $3((n-4)!)$ on the maximum number of crossings between any pair of edges. It follows that the total number of crossings is finite and upper bounded by $n!$.

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arXiv2020

Pentagon contact representations

Representations of planar triangulations as contact graphs of a set of internally disjoint homothetic triangles or of a set of internally disjoint homothetic squares have received quite some attention in recent years. In this paper we investigate representations of planar triangulations as contact graphs of a set of internally disjoint homothetic pentagons. Surprisingly such a representation exists for every triangulation whose outer face is a 5-gon. We relate these representations to five color forests. These combinatorial structures resemble Schnyder woods and transversal structures, respectively. In particular there is a bijection to certain alpha-orientations and consequently a lattice structure on the set of five color forests of a given graph. This lattice structure plays a role in an algorithm that is supposed to compute a contact representation with pentagons for a given graph. Based on a five color forest the algorithm builds a system of linear equations and solves it, if the solution is non-negative, it encodes distances between corners of a pentagon representation. In this case the representation is constructed and the algorithm terminates. Otherwise negative variables guide a change of the five color forest and the procedure is restarted with the new five color forest. Similar algorithms have been proposed for contact representations with homothetic triangles and with squares.

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arXiv2020

On the Average Complexity of the $k$-Level

Let ${\cal L}$ be an arrangement of $n$ lines in the Euclidean plane. The \emph{$k$-level} of ${\cal L}$ consists of all vertices $v$ of the arrangement which have exactly $k$ lines of ${\cal L}$ passing below $v$. The complexity (the maximum size) of the $k$-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of $O(n\cdot (k+1)^{1/3})$. Due to the correspondence between lines in the plane and great-circles on the sphere, the asymptotic bounds carry over to arrangements of great-circles on the sphere, where the $k$-level denotes the vertices at distance at most $k$ to a marked cell, the \emph{south pole}. We prove an upper bound of $O((k+1)^2)$ on the expected complexity of the $k$-level in great-circle arrangements if the south pole is chosen uniformly at random among all cells. We also consider arrangements of great $(d-1)$-spheres on the sphere $\mathbb{S}^d$ which are orthogonal to a set of random points on $\mathbb{S}^d$. In this model, we prove that the expected complexity of the $k$-level is of order $Θ((k+1)^{d-1})$.

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arXiv2020

Arrangements of Approaching Pseudo-Lines

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.

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arXiv2020

Arrangements of Pseudocircles: Triangles and Drawings

A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells $p_3$ in digon-free arrangements of $n$ pairwise intersecting pseudocircles is at least $2n-4$. We present examples to disprove this conjecture. With a recursive construction based on an example with $12$ pseudocircles and $16$ triangles we obtain a family with $p_3(\mathcal{A})/n \to 16/11 = 1.\overline{45}$. We expect that the lower bound $p_3(\mathcal{A}) \geq 4n/3$ is tight for infinitely many simple arrangements. It may however be true that all digon-free arrangements of $n$ pairwise intersecting circles have at least $2n-4$ triangles. For pairwise intersecting arrangements with digons we have a lower bound of $p_3 \geq 2n/3$, and conjecture that $p_3 \geq n-1$. Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that $p_3 \le 2n^2/3 +O(n)$. This is essentially best possible because there are families of pairwise intersecting arrangements of $n$ pseudocircles with $p_3/n^2 \to 2/3$. The paper contains many drawings of arrangements of pseudocircles and a good fraction of these drawings was produced automatically from the combinatorial data produced by our generation algorithm. In the final section we describe some aspects of the drawing algorithm.

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arXiv2020

On Covering Numbers, Young Diagrams, and the Local Dimension of Posets

We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{ř}{\'ı}k, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant local difference graph cover number and unbounded local complete bipartite cover number? We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim \emph{et al.}, we use the results on local covering numbers to provide lower and upper bounds for the local dimension of partially ordered sets of height~2. We discuss the local dimension of some posets related to Boolean lattices and show that the poset induced by the first two layers of the Boolean lattice has local dimension $(1 + o(1))\log_2\log_2 n$. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension.

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arXiv2020

Arrangements of Pseudocircles: On Circularizability

An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that any two of the curves are either disjoint or intersect in exactly two crossing points. We call an arrangement intersecting if every pair of pseudocircles intersects twice. An arrangement is circularizable if there is a combinatorially equivalent arrangement of circles. In this paper we present the results of the first thorough study of circularizability. We show that there are exactly four non-circularizable arrangements of 5 pseudocircles (one of them was known before). In the set of 2131 digon-free intersecting arrangements of 6 pseudocircles we identify the three non-circularizable examples. We also show non-circularizability of 8 additional arrangements of 6 pseudocircles which have a group of symmetries of size at least 4. Most of our non-circularizability proofs depend on incidence theorems like Miquel's. In other cases we contradict circularizability by considering a continuous deformation where the circles of an assumed circle representation grow or shrink in a controlled way. The claims that we have all non-circularizable arrangements with the given properties are based on a program that generated all arrangements up to a certain size. Given the complete lists of arrangements, we used heuristics to find circle representations. Examples where the heuristics failed were examined by hand.

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arXiv2019

Boolean dimension and tree-width

The dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if $P$ has dimension $d$, then to know whether $x \leq y$ in $P$ it is enough to check whether $x\leq y$ in each of the $d$ linear extensions of a witnessing realizer. Focusing on the encoding aspect Nešetřil and Pudlák defined a more expressive version of dimension. A poset $P$ has boolean dimension at most $d$ if it is possible to decide whether $x \leq y$ in $P$ by looking at the relative position of $x$ and $y$ in only $d$ permutations of the elements of $P$. We prove that posets with cover graphs of bounded tree-width have bounded boolean dimension. This stays in contrast with the fact that there are posets with cover graphs of tree-width three and arbitrarily large dimension. This result might be a step towards a resolution of the long-standing open problem: Do planar posets have bounded boolean dimension?

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arXiv2019

Line and Plane Cover Numbers Revisited

A measure for the visual complexity of a straight-line crossing-free drawing of a graph is the minimum number of lines needed to cover all vertices. For a given graph $G$, the minimum such number (over all drawings in dimension $d \in \{2,3\}$) is called the \emph{$d$-dimensional weak line cover number} and denoted by $π^1_d(G)$. In 3D, the minimum number of \emph{planes} needed to cover all vertices of~$G$ is denoted by $π^2_3(G)$. When edges are also required to be covered, the corresponding numbers $ρ^1_d(G)$ and $ρ^2_3(G)$ are called the \emph{(strong) line cover number} and the \emph{(strong) plane cover number}. Computing any of these cover numbers -- except $π^1_2(G)$ -- is known to be NP-hard. The complexity of computing $π^1_2(G)$ was posed as an open problem by Chaplick et al. [WADS 2017]. We show that it is NP-hard to decide, for a given planar graph~$G$, whether $π^1_2(G)=2$. We further show that the universal stacked triangulation of depth~$d$, $G_d$, has $π^1_2(G_d)=d+1$. Concerning~3D, we show that any $n$-vertex graph~$G$ with $ρ^2_3(G)=2$ has at most $5n-19$ edges, which is tight.

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arXiv2019

4-Connected Triangulations on Few Lines

We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most $\sqrt{2n}$ lines each of them horizontal or vertical. The same holds for all subgraphs of such triangulations. The proof is based on a corresponding result for diagrams of planar lattices which makes use of orthogonal chain and antichain families.

math.CO↗

arXiv2019

Complete Acyclic Colorings

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.

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arXiv2019

Drawing HV-Restricted Planar Graphs

A strict orthogonal drawing of a graph $G=(V, E)$ in $\mathbb{R}^2$ is a drawing of $G$ such that each vertex is mapped to a distinct point and each edge is mapped to a horizontal or vertical line segment. A graph $G$ is $HV$-restricted if each of its edges is assigned a horizontal or vertical orientation. A strict orthogonal drawing of an $HV$-restricted graph $G$ is good if it is planar and respects the edge orientations of $G$. In this paper, we give a polynomial-time algorithm to check whether a given $HV$-restricted plane graph (i.e., a planar graph with a fixed combinatorial embedding) admits a good orthogonal drawing preserving the input embedding, which settles an open question posed by Maňuch et al. (Graph Drawing 2010). We then examine $HV$-restricted planar graphs (i.e., when the embedding is not fixed), and give a complete characterization of the $HV$-restricted biconnected outerplanar graphs that admit good orthogonal drawings.

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