arXiv subjects

Philipp Kindermann

Publications and source records attributed to Philipp Kindermann.

At least 19 recordsLinked to original sources

arXiv2026

On the $2$-Bend Slope Number of $1$-Planar Graphs

While drawing planar graphs with few slopes and few bends is a well-studied problem, corresponding extensions to beyond-planar graphs still remain mostly unexplored. Motivated by this observation, in this work, we provide bounds on the slope number of biconnected $1$-planar graphs when two bends are allowed along each edge. Our contribution is an incremental drawing algorithm that produces $2$-bend $1$-planar drawings of biconnected $1$-plane graphs with maximum degree $Δ$ using any prescribed set of $Δ$ pairwise distinct slopes.

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arXiv2026

On the Recognition of Outerplanar Graphs with Queue Number 1

A linear layout of a graph is defined as a total order of the vertices and a partition of the edges to pages. In a stack (queue) layout, no two edges on the same page may cross (nest). The stack (queue) number of a graph is the minimum number of pages required in a stack (queue) layout. This paper focuses on characterizing and recognizing graphs that have both stack number 1 and queue number 1. It is known that the graphs with stack number 1 are exactly the outerplanar graphs. We show that (i) deciding whether a given outerplanar graph has queue number 1 is NP-hard; (ii) deciding whether a given maximal outerplanar graph has queue number 1 can be done in linear time. Moreover, we investigate the interplay between outerpaths with queue number 1 and their maximum vertex degree.

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arXiv2026

On the Complexity of Extending Storylines

Storyline layouts visualize temporal interactions by drawing each character as an x-monotone curve and enforcing that the participants of every meeting form a contiguous vertical group. We study a drawing extension variant in which a layout of a sub-storyline is fixed and has to be extended by inserting $k$ missing characters while adhering to all meeting constraints. We consider two optimization objectives: minimizing the number of additional crossings introduced to complete the storyline, and minimizing the maximum number of crossings incurred by any single character. For both variants, we analyze the parameterized complexity with respect to the natural parameter $k$ representing the size of the missing information, as well as additional structural parameters such as the number of characters $σ$ per time instant and the number of meetings $μ$ involving missing characters. We contribute a broad collection of results, most of them tight, drawing a nearly complete picture of the complexity landscape of these extension problems.

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arXiv2026

How Many Slopes Does Polynomial Area Cost?

In this work, we study the interplay between the number of slopes, the number of bends per edge, and the area requirements for planar drawings of bounded-degree graphs. Our motivation stems from the fact that, while numerous algorithms produce planar drawings with few slopes for graphs of relatively small degree in polynomial area, existing approaches for higher-degree graphs often require super-polynomial area. We address this gap in the literature by presenting new constructions that yield polynomial-area drawings with few bends per edge while slightly increasing the required number of slopes, thereby providing the first systematic study of slopes, bends and area trade-offs.

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arXiv2026

Upward-Planar Drawings with Bounded Span

We consider upward-planar layered drawings of directed graphs, i.e., crossing-free drawings in which each edge is drawn as a y-monotone curve going upward from its tail to its head, and the y-coordinates of the vertices are integers. The span of an edge in such a drawing is the absolute difference between the y-coordinates of its endpoints, and the span of the drawing is the maximum span of any edge. The span of an upward-planar graph is the minimum span over all its upward-planar drawings. We study the problem of determining the span of upward-planar graphs and provide both combinatorial and algorithmic results. On the combinatorial side, we present upper and lower bounds for the span of directed trees. On the algorithmic side, we show that the problem of determining the span of an upward-planar graph is NP-complete already for directed trees and for biconnected single-source graphs. Moreover, we give efficient algorithms for several graph families with a bounded number of sources, including st-planar graphs and graphs where the planar or upward-planar embedding is prescribed. Furthermore, we show that the problem is fixed-parameter tractable with respect to the vertex cover number and the treedepth plus the span.

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arXiv2026

Determining Factorial Speed Fast

The speed of a graph class $\cal G$ measures how many labeled graphs on $n$ vertices one can find in $\cal G$. This graph class complexity function is explicitly provided on graphclasses.org. However, for many graph classes, their speed status is classified as \emph{unknown}. In this paper, w}\shortversion{W}e show that any graph class representable by a finite binary language has at most factorial speed, meaning that its speed function behaves like $2^{Θ(n\log n)}$, and we use this criterion to classify many graph classes whose speed was previously unknown as factorial. As a consequence, inclusions between several graph classes can now be seen to be proper. We also prove that $k$-letter graphs have exponential speed, i.e., the speed function lies in $2^{Θ(n)}$.

cs.DM↗

arXiv2026

On Layered Area-Proportional Rectangle Contact Representations

Semantic word clouds visualize the semantic relatedness between the words of a text by placing pairs of related words close to each other. Formally, the problem of drawing semantic word clouds corresponds to drawing a rectangle contact representation of a graph whose vertices correlate to the words to be displayed and whose edges indicate that two words are semantically related. The goal is to maximize the number of realized contacts while avoiding any false adjacencies. We consider a variant of this problem that restricts input graphs to be layered and all rectangles to be of equal height, called \textsc{Maximum Layered Contact Representation Of Word Networks} or \textsc{Max-LayeredCrown}, as well as the variant \textsc{Max-IntLayeredCrown}, which restricts the problem to only rectangles of integer width and the placement of those rectangles to integer coordinates. We classify the corresponding decision problem $k$-\textsc{IntLayeredCrown} as NP-complete even for triangulated graphs and $k$-\textsc{LayeredCrown} as NP-complete for planar graphs. We introduce three algorithms: a 1/2-approximation for \textsc{Max-LayeredCrown} of triangulated graphs, and a PTAS and an XP algorithm for \textsc{Max-IntLayeredCrown} with rectangle width polynomial in $n$.

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arXiv2025

Parameterized Approaches to Orthogonal Compaction

Orthogonal graph drawings are used in applications such as UML diagrams, VLSI layout, cable plans, and metro maps. We focus on drawing planar graphs and assume that we are given an \emph{orthogonal representation} that describes the desired shape, but not the exact coordinates of a drawing. Our aim is to compute an orthogonal drawing on the grid that has minimum area among all grid drawings that adhere to the given orthogonal representation. This problem is called orthogonal compaction (OC) and is known to be NP-hard, even for orthogonal representations of cycles [Evans et al., 2022]. We investigate the complexity of OC with respect to several parameters. Among others, we show that OC is fixed-parameter tractable with respect to the most natural of these parameters, namely, the number of \emph{kitty corners} of the orthogonal representation: the presence of pairs of kitty corners in an orthogonal representation makes the OC problem hard. Informally speaking, a pair of kitty corners is a pair of reflex corners of a face that point at each other. Accordingly, the number of kitty corners is the number of corners that are involved in some pair of kitty corners.

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arXiv2025

Three Edge-disjoint Plane Spanning Paths in a Point Set

We consider the following problem: Given a set $S$ of $n$ distinct points in the plane, how many edge-disjoint plane straight-line spanning paths can be drawn on $S$? Each spanning path must be crossing-free, but edges from different paths are allowed to intersect at arbitrary points. It is known that if the points of $S$ are in convex position, then $\lfloor n/2 \rfloor$ such paths always exist. However, for general point sets, the best known construction yields only two edge-disjoint plane spanning paths. In this paper, we prove that for any set $S$ of at least ten points in general position (i.e., no three points are collinear), it is always possible to draw at least three edge-disjoint plane straight-line spanning paths. Our proof relies on a structural result about halving lines in point sets and builds on the known two-path construction, which we also strengthen: we show that for any set $S$ of at least six points, and for any two specified points on the boundary of the convex hull of $S$, there exist two edge-disjoint plane spanning paths that start at those prescribed points. Finally, we complement our positive results with a lower bound: for every $n \geq 6$, there exists a set of $n$ points for which no more than $\lceil n/3 \rceil$ edge-disjoint plane spanning paths are possible.

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.

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arXiv2025

Saturated Drawings of Geometric Thickness k

We investigate saturated geometric drawings of graphs with geometric thickness $k$, where no edge can be added without increasing $k$. We establish lower and upper bounds on the number of edges in such drawings if the vertices lie in convex position. We also study the more restricted version where edges are precolored, and for $k=2$ the case for vertices in non-convex position.

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arXiv2025

Outside-Obstacle Representations with All Vertices on the Outer Face

An obstacle representation of a graph $G$ consists of a set of polygonal obstacles and a drawing of $G$ as a visibility graph with respect to the obstacles: vertices are mapped to points and edges to straight-line segments such that each edge avoids all obstacles whereas each non-edge intersects at least one obstacle. Obstacle representations have been investigated quite intensely over the last few years. Here we focus on outside-obstacle representations (OORs) that use only one obstacle in the outer face of the drawing. It is known that every outerplanar graph admits such a representation. We strengthen this result by showing that every (partial) 2-tree has an OOR. We also consider restricted versions of OORs where the vertices of the graph form a convex polygon or even a regular polygon. We characterize when the complement of a tree and when a complete graph minus a simple cycle admits a convex OOR. We construct regular OORs for all (partial) outerpaths, cactus graphs, and grids.

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arXiv2024

Weakly Leveled Planarity with Bounded Span

This paper studies planar drawings of graphs in which each vertex is represented as a point along a sequence of horizontal lines, called levels, and each edge is either a horizontal segment or a strictly $y$-monotone curve. A graph is $s$-span weakly leveled planar if it admits such a drawing where the edges have span at most $s$; the span of an edge is the number of levels it touches minus one. We investigate the problem of computing $s$-span weakly leveled planar drawings from both the computational and the combinatorial perspectives. We prove the problem to be para-NP-hard with respect to its natural parameter $s$ and investigate its complexity with respect to widely used structural parameters. We show the existence of a polynomial-size kernel with respect to vertex cover number and prove that the problem is FPT when parameterized by treedepth. We also present upper and lower bounds on the span for various graph classes. Notably, we show that cycle trees, a family of $2$-outerplanar graphs generalizing Halin graphs, are $Θ(\log n)$-span weakly leveled planar and $4$-span weakly leveled planar when $3$-connected. As a byproduct of these combinatorial results, we obtain improved bounds on the edge-length ratio of the graph families under consideration.

cs.CG↗

arXiv2024

On $k$-Plane Insertion into Plane Drawings

We introduce the $k$-Plane Insertion into Plane drawing ($k$-PIP) problem: given a plane drawing of a planar graph $G$ and a set $F$ of edges, insert the edges in $F$ into the drawing such that the resulting drawing is $k$-plane. In this paper, we show that the problem is NP-complete for every $k\ge 1$, even when $G$ is biconnected and the set $F$ of edges forms a matching or a path. On the positive side, we present a linear-time algorithm for the case that $k=1$ and $G$ is a triangulation.

cs.CG↗

arXiv2024

Geometric Thickness of Multigraphs is $\exists \mathbb{R}$-complete

We say that a (multi)graph $G = (V,E)$ has geometric thickness $t$ if there exists a straight-line drawing $φ: V \rightarrow \mathbb{R}^2$ and a $t$-coloring of its edges where no two edges sharing a point in their relative interior have the same color. The \textsc{Geometric Thickness} problem asks whether a given multigraph has geometric thickness at most $t$. This problem was shown to be NP-hard for $t=2$ [Durocher, Gethner, and Mondal, CG 2016]. In this paper, we settle the computational complexity of \textsc{Geometric Thickness} by showing that it is $\exists \mathbb{R}$-complete already for thickness $30$. Moreover, our reduction shows that the problem is $\exists \mathbb{R}$-complete for $4392$-planar graphs, where a graph is $k$-planar if it admits a topological drawing with at most $k$ crossings per edge. In the course of our paper, we answer previous questions on geometric thickness and on other related problems, in particular that simultaneous graph embeddings of $31$ edge-disjoint graphs and pseudo-segment stretchability with chromatic number $30$ are $\exists \mathbb{R}$-complete.

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arXiv2024

Computing Height-Optimal Tangles Faster

We study the following combinatorial problem. Given a set of $n$ y-monotone wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset $L$ of swaps (that is, unordered pairs of numbers between 1 and $n$) and an initial order of the wires, a tangle realizes $L$ if each pair of wires changes its order exactly as many times as specified by $L$. The aim is to find a tangle that realizes $L$ using the smallest number of layers. We show that this problem is NP-hard, and we give an algorithm that computes an optimal tangle for $n$ wires and a given list $L$ of swaps in $O((2|L|/n^2+1)^{n^2/2} \cdot φ^n \cdot n)$ time, where $φ\approx 1.618$ is the golden ratio. We can treat lists where every swap occurs at most once in $O(n!φ^n)$ time. We implemented the algorithm for the general case and compared it to an existing algorithm. Finally, we discuss feasibility for lists with a simple structure.

cs.DM↗

arXiv2023

The Complexity of Finding Tangles

We study the following combinatorial problem. Given a set of $n$ y-monotone curves, which we call wires, a tangle determines the order of the wires on a number of horizontal layers such that any two consecutive layers differ only in swaps of neighboring wires. Given a multiset $L$ of swaps (that is, unordered pairs of wires) and an initial order of the wires, a tangle realizes $L$ if each pair of wires changes its order exactly as many times as specified by $L$. Deciding whether a given multiset of swaps admits a realizing tangle is known to be NP-hard [Yamanaka et al., CCCG 2018]. We prove that this problem remains NP-hard if every pair of wires swaps only a constant number of times. On the positive side, we improve the runtime of a previous exponential-time algorithm. We also show that the problem is in NP and fixed-parameter tractable with respect to the number of wires.

cs.DM↗

arXiv2023

Deciding the Feasibility and Minimizing the Height of Tangles

We study the following combinatorial problem. Given a set of $n$ y-monotone \emph{wires}, a \emph{tangle} determines the order of the wires on a number of horizontal \emph{layers} such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset~$L$ of \emph{swaps} (that is, unordered pairs of wires) and an initial order of the wires, a tangle \emph{realizes}~$L$ if each pair of wires changes its order exactly as many times as specified by~$L$. \textsc{List-Feasibility} is the problem of finding a tangle that realizes a given list~$L$ if such a tangle exists. \textsc{Tangle-Height Minimization} is the problem of finding a tangle that realizes a given list and additionally uses the minimum number of layers. \textsc{List-Feasibility} (and therefore \textsc{Tangle-Height Minimization}) is NP-hard [Yamanaka, Horiyama, Uno, Wasa; CCCG 2018]. We prove that \textsc{List-Feasibility} remains NP-hard if every pair of wires swaps only a constant number of times. On the positive side, we present an algorithm for \textsc{Tangle-Height Minimization} that computes an optimal tangle for $n$ wires and a given list~$L$ of swaps in $O((2|L|/n^2+1)^{n^2/2} \cdot φ^n \cdot n)$ time, where $φ\approx 1.618$ is the golden ratio and $|L|$ is the total number of swaps in~$L$. From this algorithm, we derive a simpler and faster version to solve \textsc{List-Feasibility}. We also use the algorithm to show that \textsc{List-Feasibility} is in NP and fixed-parameter tractable with respect to the number of wires. For \emph{simple} lists, where every swap occurs at most once, we show how to solve \textsc{Tangle-Height Minimization} in $O(n!φ^n)$ time.

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