arXiv subjects
Publications and source records attributed to Therese Biedl.
It is well-known that every planar 4-connected graph has a Hamiltonian cycle. In this paper, we study the question whether every 1-planar 4-connected graph has a Hamiltonian cycle. We show that this is false in general, even for 5-connected graphs, but true if the graph has a 1-planar drawing where every region is a triangle.
Bobbin lace is a fibre art form in which threads are braided together to form a fabric, often with a very detailed and complex design. In traditional practice, each region of the fabric is filled with a periodic texture. We establish the groundwork for non-periodic lace patterns and present three new quasiperiodic families based on Sturmian words, the Penrose tiling by thick and thin rhombs and the Ammann-bar decoration of the Penrose tiling.
It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight.
It is well-known that both the pathwidth and the outer-planarity of a graph can be used to obtain lower bounds on the height of a planar straight-line drawing of a graph. But both bounds fall short for some graphs. In this paper, we consider two other parameters, the (simple) homotopy height and the (simple) grid-major height. We discuss the relationship between them and to the other parameters, and argue that they give lower bounds on the straight-line drawing height that are never worse than the ones obtained from pathwidth and outer-planarity.
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.
Consider a set $P$ of $n$ points on the boundary of an axis-aligned square $Q$. We study the boundary-anchored packing problem on $P$ in which the goal is to find a set of interior-disjoint axis-aligned rectangles in $Q$ such that each rectangle is anchored (has a corner at some point in $P$), each point in $P$ is used to anchor at most one rectangle, and the total area of the rectangles is maximized. Here, a rectangle is anchored at a point $p$ in $P$ if one of its corners coincides with $p$. In this paper, we show how to solve this problem in time linear in $n$, provided that the points of $P$ are given in sorted order along the boundary of $Q$. We also consider the problem for anchoring squares and give an $O(n^4)$-time algorithm when the points in $P$ lie on two opposite sides of $Q$.
Following the seminal work of Erlebach and van Leeuwen in SODA 2008, we introduce the minimum ply covering problem. Given a set $P$ of points and a set $S$ of geometric objects, both in the plane, our goal is to find a subset $S'$ of $S$ that covers all points of $P$ while minimizing the maximum number of objects covering any point in the plane (not only points of $P$). For objects that are unit squares and unit disks, this problem is NP-hard and cannot be approximated by a ratio smaller than 2. We present 2-approximation algorithms for this problem with respect to unit squares and unit disks. Our algorithms run in polynomial time when the optimum objective value is bounded by a constant. Motivated by channel-assignment in wireless networks, we consider a variant of the problem where the selected unit disks must be 3-colorable, i.e., colored by three colors such that all disks of the same color are pairwise disjoint. We present a polynomial-time algorithm that achieves a 2-approximate solution, i.e., a solution that is 6-colorable. We also study the weighted version of the problem in dimension one, where $P$ and $S$ are points and weighted intervals on a line, respectively. We present an algorithm that solves this problem in $O(n + m + M )$-time where $n$ is the number of points, $m$ is the number of intervals, and $M$ is the number of pairs of overlapping intervals. This repairs a solution claimed by Nandy, Pandit, and Roy in CCCG 2017.
It is well-known that every planar graph has a Tutte path, i.e., a path $P$ such that any component of $G-P$ has at most three attachment points on $P$. However, it was only recently shown that such Tutte paths can be found in polynomial time. In this paper, we give a new proof that 3-connected planar graphs have Tutte paths, which leads to a linear-time algorithm to find Tutte paths. Furthermore, our Tutte path has special properties: it visits all exterior vertices, all components of $G-P$ have exactly three attachment points, and we can assign distinct representatives to them that are interior vertices. Finally, our running time bound is slightly stronger; we can bound it in terms of the degrees of the faces that are incident to $P$. This allows us to find some applications of Tutte paths (such as binary spanning trees and 2-walks) in linear time as well.
$Θ_6$-Graphs graphs are important geometric graphs that have many applications especially in wireless sensor networks. They are equivalent to Delaunay graphs where empty equilateral triangles take the place of empty circles. We investigate lower bounds on the size of maximum matchings in these graphs. The best known lower bound is $n/3$, where $n$ is the number of vertices of the graph. Babu et al. (2014) conjectured that any $Θ_6$-graph has a (near-)perfect matching (as is true for standard Delaunay graphs). Although this conjecture remains open, we improve the lower bound to $(3n-8)/7$. We also relate the size of maximum matchings in $Θ_6$-graphs to the minimum size of a blocking set. Every edge of a $Θ_6$-graph on point set $P$ corresponds to an empty triangle that contains the endpoints of the edge but no other point of $P$. A blocking set has at least one point in each such triangle. We prove that the size of a maximum matching is at least $β(n)/2$ where $β(n)$ is the minimum, over all theta-six graphs with $n$ vertices, of the minimum size of a blocking set. In the other direction, lower bounds on matchings can be used to prove bounds on $β$, allowing us to show that $β(n)\geq 3n/4-2$.
It is well-known that every $n$-vertex planar graph with minimum degree 3 has a matching of size at least $\frac{n}{3}$. But all proofs of this use the Tutte-Berge-formula for the size of a maximum matching. Hence these proofs are not directly algorithmic, and to find such a matching one must apply a general-purposes maximum matching algorithm, which has run-time $O(n^{1.5}α(n))$ for planar graphs. In contrast to this, this paper gives a linear-time algorithm that finds a matching of size at least $\frac{n}{3}$ in any planar graph with minimum degree 3.
A segment representation of a graph is an assignment of line segments in 2D to the vertices in such a way that two segments intersect if and only if the corresponding vertices are adjacent. Not all graphs have such segment representations, but they exist, for example, for all planar graphs. In this note, we study the resolution that can be achieved for segment representations, presuming the ends of segments must be on integer grid points. We show that any planar graph (and more generally, any graph that has a so-called $L$-representation) has a segment representation in a grid of width and height $4^n$.
These are the revised accepted papers from the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018), Barcelona, Spain, September 26 - September 28, 2018. Proceedings are also to be published by Springer in the Lecture Notes in Computer Science series.
A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence, that is increasing or decreasing, has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as a polygonal path for which every maximal sub-path, with positive- or negative-slope edges, has at least three points. Given a sequence of distinct real numbers, the rollercoaster problem asks for a maximum-length subsequence that is a rollercoaster. It was conjectured that every sequence of $n$ distinct real numbers contains a rollercoaster of length at least $\lceil n/2\rceil$ for $n>7$, while the best known lower bound is $Ω(n/\log n)$. In this paper we prove this conjecture. Our proof is constructive and implies a linear-time algorithm for computing a rollercoaster of this length. Extending the $O(n\log n)$-time algorithm for computing a longest increasing subsequence, we show how to compute a maximum-length rollercoaster within the same time bound. A maximum-length rollercoaster in a permutation of $\{1,\dots,n\}$ can be computed in $O(n \log \log n)$ time. The search for rollercoasters was motivated by orthogeodesic point-set embedding of caterpillars. A caterpillar is a tree such that deleting the leaves gives a path, called the spine. A top-view caterpillar is one of degree 4 such that the two leaves adjacent to each vertex lie on opposite sides of the spine. As an application of our result on rollercoasters, we are able to find a planar drawing of every $n$-node top-view caterpillar on every set of $\frac{25}{3}n$ points in the plane, such that each edge is an orthogonal path with one bend. This improves the previous best known upper bound on the number of required points, which is $O(n \log n)$. We also show that such a drawing can be obtained in linear time, provided that the points are given in sorted order.
The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions, like treewidth or pathwidth. In this paper, we for the first time show that crossing number is tractable (even in linear time) for maximal graphs of bounded pathwidth~3. The technique also shows that the crossing number and the rectilinear (a.k.a. straight-line) crossing number are identical for this graph class, and that we require only an $O(n)\times O(n)$-grid to achieve such a drawing. Our techniques can further be extended to devise a 2-approximation for general graphs with pathwidth 3, and a $4w^3$-approximation for maximal graphs of pathwidth $w$. This is a constant approximation for bounded pathwidth graphs.
We show that every 4-connected planar graph has a $B_3$-EPG representation, i.e., every vertex is represented by a curve on the grid with at most three bends, and two vertices are adjacent if and only if the corresponding curves share an edge of the grid. Our construction is based on a modification of the representation by touching thickened $L$-shapes proposed by Gonçalves et al.
In this paper, we study a class of graph drawings that arise from bobbin lace patterns. The drawings are periodic and require a combinatorial embedding with specific properties which we outline and demonstrate can be verified in linear time. In addition, a lace graph drawing has a topological requirement: it contains a set of non-contractible directed cycles which must be homotopic to $(1,0)$, that is, when drawn on a torus, each cycle wraps once around the minor meridian axis and zero times around the major longitude axis. We provide an algorithm for finding the two fundamental cycles of a canonical rectangular schema in a supergraph that enforces this topological constraint. The polygonal schema is then used to produce a straight-line drawing of the lace graph inside a rectangular frame. We argue that such a polygonal schema always exists for combinatorial embeddings satisfying the conditions of bobbin lace patterns, and that we can therefore create a pattern, given a graph with a fixed combinatorial embedding of genus one.
Let $T$ be an $n$-node tree of maximum degree 4, and let $P$ be a set of $n$ points in the plane with no two points on the same horizontal or vertical line. It is an open question whether $T$ always has a planar drawing on $P$ such that each edge is drawn as an orthogonal path with one bend (an "L-shaped" edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set $P$ for which such drawings are possible to: $O(n^{1.55})$ for maximum degree 4 trees; $O(n^{1.22})$ for maximum degree 3 (binary) trees; and $O(n^{1.142})$ for perfect binary trees. Drawing ordered trees with L-shaped edges is harder---we give an example that cannot be done and a bound of $O(n \log n)$ points for L-shaped drawings of ordered caterpillars, which contrasts with the known linear bound for unordered caterpillars.
In an EPG-representation of a graph $G$ each vertex is represented by a path in the rectangular grid, and $(v,w)$ is an edge in $G$ if and only if the paths representing $v$ an $w$ share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x- and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters. We show that there are $m$-edge graphs that require a grid of area $Ω(m)$ in any variant of EPG-representations. Similarly there are pathwidth-$k$ graphs that require height $Ω(k)$ and area $Ω(kn)$ in any variant of EPG-representations. We prove a matching upper bound of $O(kn)$ area for all pathwidth-$k$ graphs in the strongest model, the one where edges are required to be both x- and y-monotone. Thus in this strongest model, the result implies, for example, $O(n)$, $O(n \log n)$ and $O(n^{3/2})$ area bounds for bounded pathwidth graphs, bounded treewidth graphs and all classes of graphs that exclude a fixed minor, respectively. For the model with no restrictions on the monotonicity of the edges, stronger results can be achieved for some graph classes, for example an $O(n)$ area bound for bounded treewidth graphs and $O(n \log^2 n)$ bound for graphs of bounded genus.