arXiv subjects

Therese Biedl

Publications and source records attributed to Therese Biedl.

At least 19 recordsLinked to original sources

arXiv2026

On Reconstructing a Convex Polygon from Partial Information

The reconstruction problem asks to construct a (convex) polygon that has a specified set of features, such as an ordered set of edge-lengths or an ordered set of polygon-angles. In this paper, we do a systematic exploration of the reconstruction problem in all scenarios where one or two sets of features have been specified. Some of these scenarios were well-studied already, for some we develop testing-algorithms and/or hardness results, and many give rise to interesting open problems for future study.

cs.CG↗

arXiv2026

Unbent collections of non-planar $s$-grid-drawing

In a recent paper, Anti\'{c} et al.~studied collections of planar orthogonal drawings of a graph where every edge is unbent in at least one drawing. This paper generalizes this concept to non-planar drawings, and shows that then two drawings always suffice (for planar drawings three drawings are sometimes needed). The results can also be generalized to $s$-grid drawings for $s\geq 3$.

cs.CG↗

arXiv2026

The Twin-Width of Graphs of Bounded VC-Dimension

In this paper, we investigate which hereditary classes of graphs admit sub-linear (in the number of vertices) bounds on twin-width. By modifying conference graphs, we can show that split, bipartite, and co-bipartite graphs can all have linear twin-width. However, excluding an induced subgraph of each of these three types is equivalent to the class of graphs having bounded VC-dimension, as shown by Bousquet, Lagoutte, Li, Parreau and Thomass\'e. Graphs of bounded VC-dimension can have unbounded twin-width, but whether it can be linear was an open question. In this paper, we first present a tool for obtaining twin-width bounds in general by contracting a graph based on a partition by distinct neighbourhoods. Then, using this tool, we prove that graphs with bounded VC-dimension have twin-width at most sub-linear. We also obtain a separate, tighter upper bound for the class of interval graphs, as well as a lower bound.

math.CO↗

arXiv2026

Bipartite Graphs Are Not Well-Quasi-Ordered by Bipartite Minors

In "Bipartite minors" [Journal of Combinatorial Theory, Series B, 2016], Chudnovsky et al. introduced the bipartite minor relation, a quasi-order on the class of bipartite graphs somewhat analogous the minor relation on general graphs and asked whether it is a well-quasi-order. We answer this question negatively by giving an infinite set of 2-connected bipartite graphs that are pairwise incomparable with respect to the bipartite minor relation. We additionally give two sets of infinitely many pairs of bipartite graphs: one set of pairs G, H such that H is a bipartite minor, but not a minor, of G, and one set of pairs G, H such that H is a minor, but not a bipartite minor, of G.

math.CO↗

arXiv2026

Using Ray-shooting Queries for Sublinear Algorithms for Dominating Sets in RDV Graphs

In this paper, we study the dominating set problem in \emph{RDV graphs}, a graph class that lies between interval graphs and chordal graphs and is defined as the \textbf{v}ertex-intersection graphs of \textbf{d}ownward paths in a \textbf{r}ooted tree. It was shown in a previous paper that adjacency queries in an RDV graph can be reduced to the question whether a horizontal segment intersects a vertical segment. This was then used to find a maximum matching in an $n$-vertex RDV graph, using priority search trees, in $O(n\log n)$ time, i.e., without even looking at all edges. In this paper, we show that if additionally we also use a ray shooting data structure, we can also find a minimum dominating set in an RDV graph $O(n\log n)$ time (presuming a linear-sized representation of the graph is given). The same idea can also be used for a new proof to find a minimum dominating set in an interval graph in $O(n)$ time.

cs.DS↗

arXiv2025

Face-hitting dominating sets in planar graphs: Alternative proof and linear-time algorithm

In a recent paper, Francis, Illickan, Jose and Rajendraprasad showed that every $n$-vertex plane graph $G$ has (under some natural restrictions) a vertex-partition into two sets $V_1$ and $V_2$ such that each $V_i$ is \emph{dominating} (every vertex of $G$ contains a vertex of $V_i$ in its closed neighbourhood) and \emph{face-hitting} (every face of $G$ is incident to a vertex of $V_i$). Their proof works by considering a supergraph $G'$ of $G$ that has certain properties, and among all such graphs, taking one that has the fewest edges. As such, their proof is not algorithmic. Their proof also relies on the 4-color theorem, for which a quadratic-time algorithm exists, but it would not be easy to implement. In this paper, we give a new proof that every $n$-vertex plane graph $G$ has (under the same restrictions) a vertex-partition into two dominating face-hitting sets. Our proof is constructive, and requires nothing more complicated than splitting a graph into 2-connected components, finding an ear decomposition, and computing a perfect matching in a 3-regular plane graph. For all these problems, linear-time algorithms are known and so we can find the vertex-partition in linear time.

cs.DS↗

arXiv2025

Sweeping $x$-monotone pseudolines

We study the problem of sweeping a pseudoline arrangement with $n$ $x$-monotone curves with a rope (an $x$-monotone curve that connects the points at infinity). The rope can move by flipping over a face of the arrangement, replacing parts of it from the lower to the upper chain of the face. Counting as length of the rope the number of edges, what rope-length can be needed in such a sweep? We show that all such arrangements can be swept with rope-length at most $2n-2$, and for some arrangements rope-length at least $7(n-2)/4+1$ is required. We also discuss some complexity issues around the problem of computing a sweep with the shortest rope-length.

cs.CG↗

arXiv2025

Rectangular Duals on the Cylinder and the Torus

A rectangular dual of a plane graph $G$ is a contact representation of $G$ by interior-disjoint rectangles such that (i) no four rectangles share a point, and (ii) the union of all rectangles is a rectangle. In this paper, we study rectangular duals of graphs that are embedded in surfaces other than the plane. In particular, we fully characterize when a graph embedded on a cylinder admits a cylindrical rectangular dual. For graphs embedded on the flat torus, we can test whether the graph has a toroidal rectangular dual if we are additionally given a \textit{regular edge labeling}, i.e. a combinatorial description of rectangle adjacencies. Furthermore we can test whether there exists a toroidal rectangular dual that respects the embedding and that resides on a flat torus for which the sides are axis-aligned. Testing and constructing the rectangular dual, if applicable, can be done efficiently.

cs.CG↗

arXiv2025

Finding a Shortest Curve that Separates Few Objects from Many

We present a fixed-parameter tractable (FPT) algorithm to find a shortest curve that encloses a set of k required objects in the plane while paying a penalty for enclosing unwanted objects. The input is a set of interior-disjoint simple polygons in the plane, where k of the polygons are required to be enclosed and the remaining optional polygons have non-negative penalties. The goal is to find a closed curve that is disjoint from the polygon interiors and encloses the k required polygons, while minimizing the length of the curve plus the penalties of the enclosed optional polygons. If the penalties are high, the output is a shortest curve that separates the required polygons from the others. The problem is NP-hard if k is not fixed, even in very special cases. The runtime of our algorithm is $O(3^kn^3)$, where n is the number of vertices of the input polygons. We extend the result to a graph version of the problem where the input is a connected plane graph with positive edge weights. There are k required faces; the remaining faces are optional and have non-negative penalties. The goal is to find a closed walk in the graph that encloses the k required faces, while minimizing the weight of the walk plus the penalties of the enclosed optional faces. We also consider an inverted version of the problem where the required objects must lie outside the curve. Our algorithms solve some other well-studied problems, such as geometric knapsack.

cs.CG↗

arXiv2025

Strongly chordal graphs as intersection graphs of trees (Farber's proof revisited)

In his Ph.D. thesis, Farber proved that every strongly chordal graph can be represented as intersection graph of subtrees of a weighted tree, and these subtrees are ``compatible''. Moreover, this is an equivalent characterization of strongly chordal graphs. To my knowledge, Farber never published his results in a conference or a journal, and the thesis is not available electronically. As a service to the community, I therefore reproduce the proof here. I then answer some questions that naturally arise from the proof. In particular, the sufficiency proof works by showing the existence of a simple vertex. I give here an alternate sufficiency proof that directly converts a set of compatible subtrees into a strong elimination order.

math.CO↗

arXiv2024

The basis number of 1-planar graphs

Let $B$ be a set of Eulerian subgraphs of a graph $G$. We say $B$ forms a $k$-basis if it is a minimum set that generates the cycle space of $G$, and any edge of $G$ lies in at most $k$ members of $B$. The basis number of a graph $G$, denoted by $b(G)$, is the smallest integer such that $G$ has a $k$-basis. A graph is called 1-planar (resp. planar) if it can be embedded in the plane with at most one crossing (resp. no crossing) per edge. MacLane's planarity criterion characterizes planar graphs based on their cycle space, stating that a graph is planar if and only if it has a $2$-basis. We study here the basis number of 1-planar graphs, demonstrate that it is unbounded in general, and show that it is bounded for many subclasses of 1-planar graphs.

math.CO↗

arXiv2024

Computing Conforming Partitions with Low Stabbing Number for Rectilinear Polygons

A conforming partition of a rectilinear n-gon P (possibly with holes) is a partition of P into rectangles without using Steiner points (i.e., all corners of all rectangles must lie on the boundary of P). The stabbing number of such a partition is the maximum number of rectangles intersected by an axis-aligned segment lying in the interior of P. In this paper, we examine the problem of computing conforming partitions with low stabbing number. We show that computing a conforming partition with stabbing number at most 4 is NP-hard, which strengthens a previously known hardness result [Durocher \& Mehrabi, Theor. Comput. Sci. 689: 157-168 (2017)] and eliminates the possibility for fixed-parameter-tractable algorithms parameterized by the stabbing number unless P = NP. In contrast, we give (i) an O(n log n)-time algorithm to decide whether a conforming partition with stabbing number 2 exists, (ii) a fixed-parameter-tractable algorithm parameterized by both the stabbing number and treewidth of the pixel graph of the polygon, and (iii) a fixed-parameter-tractable algorithm parameterized by the stabbing number for polygons without holes in general position.

cs.CG↗

arXiv2024

On the $d$-independence number in 1-planar graphs

The $d$-independence number of a graph $G$ is the largest possible size of an independent set $I$ in $G$ where each vertex of $I$ has degree at least $d$ in $G$. Upper bounds for the $d$-independence number in planar graphs are well-known for $d=3,4,5$, and can in fact be matched with constructions that actually have minimum degree $d$. In this paper, we explore the same questions for 1-planar graphs, i.e., graphs that can be drawn in the plane with at most one crossing per edge. We give upper bounds for the $d$-independence number for all $d$. Then we give constructions that match the upper bound, and (for small $d$) also have minimum degree $d$.

math.CO↗

arXiv2024

Morphing Planar Graph Drawings via Orthogonal Box Drawings

We give an algorithm to morph planar graph drawings that achieves small grid size at the expense of allowing a constant number of bends on each edge. The input is an $n$-vertex planar graph and two planar straight-line drawings of the graph on an $O(n) \times O(n)$ grid. The planarity-preserving morph is composed of $O(n)$ linear morphs between successive pairs of drawings, each on an $O(n) \times O(n)$ grid with a constant number of bends per edge. The algorithm to compute the morph runs in $O(n^2)$ time on a word RAM model with standard arithmetic operations -- in particular no square roots or cube roots are required. The first step of the algorithm is to morph each input drawing to a planar orthogonal box drawing where vertices are represented by boxes and each edge is drawn as a horizontal or vertical segment. The second step is to morph between planar orthogonal box drawings. This is done by extending known techniques for morphing planar orthogonal drawings with point vertices.

cs.CG↗

arXiv2024

The Price of Upwardness

Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward $k$-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most $k$ times for some integer $k \ge 1$. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that testing upward-$k$-planarity is NP-complete already for $k=1$ and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward 1-planar drawing in which all vertices are incident to the outer face.

cs.CG↗

arXiv2024

Improved Outerplanarity Bounds for Planar Graphs

In this paper, we study the outerplanarity of planar graphs, i.e., the number of times that we must (in a planar embedding that we can initially freely choose) remove the outerface vertices until the graph is empty. It is well-known that there are $n$-vertex graphs with outerplanarity $\tfrac{n}{6}+Θ(1)$, and not difficult to show that the outerplanarity can never be bigger. We give here improved bounds of the form $\tfrac{n}{2g}+2g+O(1)$, where $g$ is the fence-girth, i.e., the length of the shortest cycle with vertices on both sides. This parameter $g$ is at least the connectivity of the graph, and often bigger; for example, our results imply that planar bipartite graphs have outerplanarity $\tfrac{n}{8}+O(1)$. We also show that the outerplanarity of a planar graph $G$ is at most $\tfrac{1}{2}$diam$(G)+O(\sqrt{n})$, where diam$(G)$ is the diameter of the graph. All our bounds are tight up to smaller-order terms, and a planar embedding that achieves the outerplanarity bound can be found in linear time.

cs.DS↗

arXiv2024

Computing Vertex and Edge Connectivity of Graphs Embedded with Crossings

Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs embedded on the plane with crossings. For planar graphs -- which can be embedded on the plane without any crossings -- it has long been known that vertex and edge connectivity can be computed in linear time. Recently, the algorithm for vertex connectivity was extended from planar graphs to 1-plane graphs (where each edge is crossed at most once) without $\times$-crossings -- these are crossings whose endpoints induce a matching. The key insight, for both these classes of graphs, is that any two vertices/edges of a minimum vertex/edge cut have small face-distance (distance measured by number of faces) in the embedding. In this paper, we attempt at a comprehensive generalization of this idea to a wider class of graphs embedded on the plane. Our method works for all those embedded graphs where every pair of crossing edges is connected by a path whose vertices and edges have a small face-distance from the crossing point. Important examples of such graphs include optimal 2-planar and optimal 3-planar graphs, $d$-map graphs, $d$-framed graphs, graphs with bounded crossing number, and $k$-plane graphs with bounded number of $\times$-crossings. For all these graph classes, we get a linear-time algorithm for computing vertex and edge connectivity.

cs.DS↗

arXiv2024

Finding maximum matchings in RDV graphs efficiently

In this paper, we study the maximum matching problem in RDV graphs, i.e., graphs that are vertex-intersection graphs of downward paths in a rooted tree. We show that this problem can be reduced to a problem of testing (repeatedly) whether a vertical segment intersects one of a dynamically changing set of horizontal segments, which in turn reduces to an orthogonal ray shooting query. Using a suitable data structure, we can therefore find a maximum matching in $O(n\log n)$ time (presuming a linear-sized representation of the graph is given), i.e., without even looking at all edges.

cs.CG↗