arXiv subjects

Johannes Zink

Publications and source records attributed to Johannes Zink.

At least 19 recordsLinked to original sources

arXiv2026

Simultaneous Embedding of Two Paths on the Grid

Simultaneous geometric embedding of graphs on an integer grid is a classical research topic in Graph Drawing. When the graphs are two paths, one can additionally require that one path is x-monotone and the other is y-monotone (the monotone setting). A simple and elegant algorithm is known to simultaneously embed two paths on an integer grid in the monotone setting. The resulting embedding is, however, not optimal in terms of grid size or edge lengths. In this paper, we study some quality measures of simultaneous embeddings of two paths on an integer grid, in both the monotone and in the unconstrained setting. In the monotone setting, we present almost-linear time algorithms to minimize either the perimeter of the bounding box or the sum of the edge lengths in the $L_1$-norm. Both results are based on transforming the problem into that of finding a minimum vertex cover (weighted for the $L_1$-norm problem) on a related bipartite graph derived from the two paths. As for the unconstrained setting, we show that both minimizing the length of the longest edge and minimizing the sum of the edge lengths of a simultaneous embedding of two paths on an integer grid is NP-hard.

cs.CG↗

arXiv2026

Embedding Trees and Cactus Graphs with Integer Edge Lengths on a Compact Integer Grid

Every planar graph can be embedded with crossing-free, straight-line edges on an $O(n) \times O(n)$ integer grid, where $n$ is the number of vertices. It is not known, however, whether integer edge lengths can be guaranteed for all planar graphs (conjectured to be true by Harborth). Kleber conjectures something even stronger: Every planar graph admits a planar straight-line grid drawing where every edge has integer length. Positive results for Kleber's conjecture are known for several classes of planar graphs, however, all but one of the existing results are existential and do not provide bounds on the size of the integer grid. We address this gap for trees and cactus graphs. For trees, we present a linear-time algorithm that computes a crossing-free straight-line drawing with integer edge lengths on an $O(n \log n) \times O(n \log n)$ grid. We complement this result by giving an $Ω(n^2)$ lower bound. For cactus graphs, we present a linear-time algorithm that computes a crossing-free straight-line drawing with integer edge lengths on an $O(n^2 \log n) \times O(n^2 \log n)$ grid.

cs.CG↗

arXiv2026

Outerplanar and Forest Storyplans

We study the problem of gradually representing a complex graph as a sequence of drawings of small subgraphs whose union is the complex graph. The sequence of drawings is called \emph{storyplan}, and each drawing in the sequence is called a \emph{frame}. In an (outer)planar storyplan, every frame is (outer)planar; in a forest storyplan, every frame is acyclic. Binucci, Di Giacomo, Lenhart, Liotta, Montecchiani, Nöllenburg, and Symvonis [JCSS 2024] proved that every graph of treewidth at most 3 admits a planar storyplan and that deciding whether a given graph admits a \emph{planar} storyplan is NP-complete. They also presented two FPT algorithms, one parameterized with respect to the vertex cover number and one with respect to the feedback edge set number of the input graph. We first prove that deciding whether a given graph admits an outerplanar storyplan (or a forest storyplan) is NP-complete. Then, we show that the FPT algorithms of Binucci et al. also work for our problem variants with small modifications. We identify graph families that admit outerplanar and forest storyplans and graph families for which such storyplans do not always exist. In the affirmative case, we present efficient algorithms that produce storyplans with straight-line edges.

cs.CG↗

arXiv2026

Using Reinforcement Learning to Optimize the Global and Local Crossing Number

Graph drawing concerns the algorithmic visualization of graphs. A good drawing of a graph is easy to read and facilitates solving tasks on the graph. Several properties have been identified to occur in good drawings of graphs. Such properties include a low number of crossings, large angles between edges, short edges, and depicting symmetries. Many of these properties are explicitly measurable metrics. This lets us model a graph-drawing problem as a game where a single player iteratively moves vertices of a straight-line graph drawing to reduce edge crossings. We investigate whether reinforcement learning can discover effective strategies for playing this game. Our reinforcement-learning agent observes the local geometric and structural context of a vertex and selects a movement direction with the goal of reducing either the global or the local crossing number, that is, either the total number of crossings or the maximum number of crossings per edge. We compare the resulting strategies to existing methods and established crossing-minimization heuristics on standard benchmark graphs. While our approach does not out-compete state-of-the-art methods for minimizing the global crossing number, it is competitive and often superior for minimizing the local crossing number.

cs.CG↗

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.

cs.CG↗

arXiv2026

Weighted Book Thickness

We introduce and study the weighted book thickness of graphs. A $k$-page book embedding of a graph $G=(V,E)$ is defined by a spanning cycle $C$ for $V$ (which does not need to be part of $G$) and a partition $E=\bigcup_{i=1}^{k}E_i$ such that $E\cap C\subseteq E_1$ and each graph $G_i=(V,E_i\cup C)$, for $1 \le i \le k$, is outerplane with outer cycle $C$. If $e\in E_i$, we say that $e$ appears on Page $i$. The classical book thickness of a graph $G$ is the minimum $k$ such that there exists a $k$-page book embedding of $G$, that is, the minimum (over all book embeddings of $G$) achievable maximum page an edge appears on. In contrast, the weighted book thickness is the minimum achievable average page an edge appears on. The embeddings that realize weighted book thickness can differ from those that realize (classical) book thickness. We show that, although every planar graph on at most nine vertices admits a 2-page book embedding realizing its weighted book thickness, already for ten vertices, there is a planar graph for which every realization of its weighted book thickness needs more pages than its book thickness. We prove that there even exists a 2-tree whose weighted book thickness cannot be realized on two pages. On the positive side, we show that for every graph of pathwidth at most two, the weighted book thickness can always be realized by a 2-page book embedding and such an embedding can be found in linear time. Moreover, we prove that it is NP-complete to decide if the weighted book thickness is at most $k$, for some given integer $k$.

cs.CG↗

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.

cs.CG↗

arXiv2026

Exploring MLLMs Perception of Network Visualization Principles

In this paper, we test whether Multimodal Large Language Models (MLLMs) can match human-subject performance in tasks involving the perception of properties in network layouts. Specifically, we replicate a human-subject experiment about perceiving quality (namely stress) in network layouts using GPT-4o, Gemini-2.5 and Qwen2.5. Our experiments show that giving MLLMs the same study information as trained human participants yields performance comparable to that of human experts and exceeds that of untrained non-experts. Additionally, we show that prompt engineering that deviates from the human-subject experiment can lead to better-than-human performance in some settings. Interestingly, like human subjects, the MLLMs seem to rely on visual proxies rather than computing the actual value of stress, indicating some sense or facsimile of perception. Explanations from the models are similar to those used by the human participants (e.g., an even distribution of nodes and uniform edge lengths).

cs.HC↗

arXiv2026

Morphing Graph Drawings in the Presence of Point Obstacles

A crossing-free morph is a continuous deformation between two graph drawings that preserves straight-line pairwise noncrossing edges. Motivated by applications in 3D morphing problems, we initiate the study of morphing graph drawings in the plane in the presence of stationary point obstacles, which need to be avoided throughout the deformation. As our main result, we prove that it is NP-hard to decide whether such an obstacle-avoiding 2D morph between two given drawings of the same graph exists. In fact, this statement remains true even in the severely restricted special case where only three vertices have to change positions. This is in sharp contrast to the classical case without obstacles, where there is an efficiently verifiable (necessary and sufficient) criterion for the existence of a morph. Further, we provide several combinatorial results related to conditions under which the existence of a morph between two drawings of a graph can or cannot be prevented by the placement of a given number of point obstacles.

cs.CG↗

arXiv2025

Edge-Constrained Hamiltonian Paths on a Point Set

Let S be a set of distinct points in general position in the Euclidean plane. A plane Hamiltonian path on S is a crossing-free geometric path such that every point of S is a vertex of the path. It is known that, if S is sufficiently large, there exist three edge-disjoint plane Hamiltonian paths on S. In this paper we study an edge-constrained version of the problem of finding Hamiltonian paths on a point set. We first consider the problem of finding a single plane Hamiltonian path pi with endpoints s, t in S and constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) ab in pi; (ii) ab not in pi. We characterize those quintuples (S, a, b, s, t) for which pi exists. Secondly, we consider the problem of finding two plane Hamiltonian paths pi_1, pi_2 on a set S with constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) pi_1 and pi_2 share no edges and ab is an edge of pi_1; (ii) pi_1 and pi_2 share no edges and none of them includes ab as an edge; (iii) both pi_1 and pi_2 include ab as an edge and share no other edges. In all cases, we characterize those triples (S, a, b) for which pi_1 and pi_2 exist.

cs.CG↗

arXiv2025

Hypergraphs as Metro Maps: Drawing Paths with Few Bends in Trees, Cacti, and Plane 4-Graphs

A hypergraph consists of a set of vertices and a set of subsets of vertices, called hyperedges. In the metro map metaphor, each hyperedge is represented by a path (the metro line) and the union of all these paths is the support graph (metro network) of the hypergraph. Formally speaking, a path-based support is a graph together with a set of paths. We consider the problem of constructing drawings of path-based supports that (i) minimize the sum of the number of bends on all paths, (ii) minimize the maximum number of bends on any path, or (iii) maximize the number of 0-bend paths, then the number of 1-bend paths, etc. We concentrate on straight-line drawings of path-based tree and cactus supports as well as orthogonal drawings of path-based plane supports with maximum degree 4.

cs.CG↗

arXiv2025

Linear Layouts of Graphs with Priority Queues

A linear layout of a graph consists of a linear ordering of its vertices and a partition of its edges into pages such that the edges assigned to the same page obey some constraint. The two most prominent and widely studied types of linear layouts are stack and queue layouts, in which any two edges assigned to the same page are forbidden to cross and nest, respectively. The names of these two layouts derive from the fact that, when parsing the graph according to the linear vertex ordering, the edges in a single page can be stored using a single stack or queue, respectively. Recently, the concepts of stack and queue layouts have been extended by using a double-ended queue or a restricted-input queue for storing the edges of a page. We extend this line of study to edge-weighted graphs by introducing priority queue layouts, that is, the edges on each page are stored in a priority queue whose keys are the edge weights. First, we show that there are edge-weighted graphs that require a linear number of priority queues. Second, we characterize the graphs that admit a priority queue layout with a single queue, regardless of the edge-weight function, and we provide an efficient recognition algorithm. Third, we show that the number of priority queues required independently of the edge-weight function is bounded by the pathwidth of the graph, but can be arbitrarily large already for graphs of treewidth two. Finally, we prove that determining the minimum number of priority queues is NP-complete if the linear ordering of the vertices is fixed.

cs.DM↗

arXiv2025

Unbent Collections of Orthogonal Drawings

Recently, there has been interest in representing single graphs by multiple drawings; for example, using graph stories, storyplans, or uncrossed collections. In this paper, we apply this idea to orthogonal graph drawing. Due to the orthogonal drawing style, we focus on 4-graphs, that is, graphs of maximum degree 4. We restrict ourselves to plane graphs, that is, planar graphs whose embedding is fixed. Our goal is to represent any plane 4-graph $G$ by an unbent collection, that is, a collection of orthogonal drawings of $G$ that adhere to the embedding of $G$ and ensure that each edge of $G$ is drawn without bends in at least one of the drawings. We investigate two objectives. First, we consider minimizing the number of drawings in an unbent collection. We prove that every plane 4-graph can be represented by a collection with at most three drawings, which is tight. We also give necessary and sufficient conditions for a graph to admit an unbent collection of size $2$. Second, we consider minimizing the total number of bends over all drawings in an unbent collection. We show that this problem is NP-hard and give a 3-approximation algorithm. For the special case of plane triconnected cubic graphs, we show how to compute minimum-bend collections in linear time.

cs.CG↗

arXiv2024

Constrained and Ordered Level Planarity Parameterized by the Number of Levels

The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant Constrained Level Planarity (CLP), each level $y$ is equipped with a partial order $\prec_y$ on its vertices and in the desired drawing the left-to-right order of vertices on level $y$ has to be a linear extension of $\prec_y$. Ordered Level Planarity (OLP) corresponds to the special case of CLP where the given partial orders $\prec_y$ are total orders. Previous results by Brückner and Rutter [SODA 2017] and Klemz and Rote [ACM Trans. Alg. 2019] state that both CLP and OLP are NP-hard even in severely restricted cases. In particular, they remain NP-hard even when restricted to instances whose width (the maximum number of vertices that may share a common level) is at most two. In this paper, we focus on the other dimension: we study the parameterized complexity of CLP and OLP with respect to the height (the number of levels). We show that OLP parameterized by the height is complete with respect to the complexity class XNLP, which was first studied by Elberfeld et al. [Algorithmica 2015] (under a different name) and recently made more prominent by Bodlaender et al. [FOCS 2021]. It contains all parameterized problems that can be solved nondeterministically in time $f(k) n^{O(1)}$ and space $f(k) \log n$ (where $f$ is a computable function, $n$ is the input size, and $k$ is the parameter). If a problem is XNLP-complete, it lies in XP, but is W[$t$]-hard for every $t$. In contrast to the fact that OLP parameterized by the height lies in XP, it turns out that CLP is NP-hard even when restricted to instances of height 4. We complement this result by showing that CLP can be solved in polynomial time for instances of height at most 3.

cs.CG↗

arXiv2024

Simultaneous Drawing of Layered Trees

We study the crossing-minimization problem in a layered graph drawing of planar-embedded rooted trees whose leaves have a given total order on the first layer, which adheres to the embedding of each individual tree. The task is then to permute the vertices on the other layers (respecting the given tree embeddings) in order to minimize the number of crossings. While this problem is known to be NP-hard for multiple trees even on just two layers, we describe a dynamic program running in polynomial time for the restricted case of two trees. If there are more than two trees, we restrict the number of layers to three, which allows for a reduction to a shortest-path problem. This way, we achieve XP-time in the number of trees.

cs.DM↗

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.

cs.CG↗