Search

The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer $k$-planar graphs, that is, graphs that admit a straight-line drawing where all the vertices lie on a circle, and every edge is crossed by at most $k$ other edges. Wood and Telle [New York J. Math., 2007] showed that every outer $k$-planar graph has treewidth at most $3k + 11$ using so-called planar decompositions, and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer $1$-planar graphs is at most $3$, which is tight. In this paper, we improve the general upper bound to $1.5k + 2$ and give a tight bound of $4$ for $k = 2$. We also establish a lower bound: we show that, for every even $k$, there is an outer $k$-planar graph with treewidth $k+2$. Our new bound immediately implies a better bound on the cop number, which answers an open question of Durocher et al. [GD 2023] in the affirmative. Our treewidth bound relies on a new and simple triangulation method for outer $k$-planar graphs that yields few crossings with graph edges per edge of the triangulation. Our method also enables us to obtain a tight upper bound of $k + 2$ for the separation number of outer $k$-planar graphs, improving an upper bound of $2k + 3$ by Chaplick et al. [GD 2017]. We also consider outer min-$k$-planar graphs, a generalization of outer $k$-planar graphs, where we achieve smaller improvements., Comment: Appears in the Proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

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 $O((2|L|/n^2+1)^{n^2/2} \cdot \varphi^n \cdot n)$ time, where $\varphi \approx 1.618$ 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 $O(n!\varphi^n)$ time., Comment: This work is a merger of arXiv:1901.06548 and arXiv:2002.12251

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. 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., Comment: Appears in Proc. SOFSEM 2024

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 outerplanar storyplan, every frame is outerplanar; in a forest storyplan, every frame is acyclic. We identify graph families that admit such storyplans and families for which such storyplans do not always exist. In the affirmative case, we present efficient algorithms that produce straight-line storyplans., Comment: Appears in Proc. SOFSEM 2024

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 [Alpert, Koch, Laison; DCG 2010]. 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 lie on a convex polygon or 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., Comment: Appears in the Proceedings of the 30th International Symposium on Graph Drawing and Network Visualization (GD 2022)

We propose a new representation of $k$-partite, $k$-uniform hypergraphs, that is, a hypergraph with a partition of vertices into $k$ parts such that each hyperedge contains exactly one vertex of each type; we call them $k$-hypergraphs for short. Given positive integers $\ell, d$, and $k$ with $\ell\leq d-1$ and $k={d\choose\ell}$, any finite set $P$ of points in $\mathbb{R}^d$ represents a $k$-hypergraph $G_P$ as follows. Each point in $P$ is covered by $k$ many axis-aligned affine $\ell$-dimensional subspaces of $\mathbb{R}^d$, which we call $\ell$-subspaces for brevity and which form the vertex set of $G_P$. We interpret each point in $P$ as a hyperedge of $G_P$ that contains each of the covering $\ell$-subspaces as a vertex. The class of \emph{$(d,\ell)$-hypergraphs} is the class of $k$-hypergraphs that can be represented in this way. The resulting classes of hypergraphs are fairly rich: Every $k$-hypergraph is a $(k,k-1)$-hypergraph. On the other hand, $(d,\ell)$-hypergraphs form a proper subclass of the class of all $k$-hypergraphs for $\ell

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., Comment: This paper has been superseded by arXiv:2312.16213 (merged from arXiv:1901.06548 and this paper). This paper has appeared in Proceedings of the 48th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2023): https://doi.org/10.1007/978-3-031-23101-8_1

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 $O((2|L|/n^2+1)^{n^2/2} \cdot \varphi^n \cdot n)$ time, where $\varphi \approx 1.618$ is the golden ratio. We can treat lists where every swap occurs at most once in $O(n!\varphi^n)$ 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., Comment: Except for its experimental Section 4, this paper has been superseded by arXiv:2312.16213 (merged from arXiv:2002.12251 and this paper). This paper has appeared in Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019)