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We introduce and study level-planar straight-line drawings with a fixed number $\lambda$ of slopes. For proper level graphs, we give an $O(n \log^2 n / \log \log n)$-time algorithm that either finds such a drawing or determines that no such drawing exists. Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks about the existence of drawings of two graphs whose restrictions to their shared subgraph coincide. We present $O(n^{4/3} \log n)$-time and $O({\lambda} n^{10/3} \log n)$-time algorithms for these respective problems on proper level-planar graphs. We complement these positive results by showing that testing whether non-proper level graphs admit level-planar drawings with $\lambda$ slopes is $\textsf{NP}$-hard even in restricted cases., Comment: Appears in the Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019)

A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and of a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. Recently, Pupyrev disproved this conjectured by demonstrating a planar partial 3-tree that does not admit a 1-stack 1-queue layout. In this note, we strengthen Pupyrev’s result by showing that the conjecture does not hold even for 2-trees, also known as series-parallel graphs.

We introduce and study level-planar straight-line drawings with a fixed number \(\lambda \) of slopes. For proper level graphs, we give an \(O(n \log ^2 n / \log \log n)\)-time algorithm that either finds such a drawing or determines that no such drawing exists. Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks about the existence of drawings of two graphs whose restrictions to their shared subgraph coincide. We present \(O(n^{4/3} \log n)\)-time and \(O(\lambda n^{10/3} \log n)\)-time algorithms for these respective problems on proper level-planar graphs.

Let \(G=(V, E)\) be a planar graph and let \(\mathcal V\) be a partition of V. We refer to the graphs induced by the vertex sets in Open image in new window as clusters. Let \(\mathcal {D}_{\mathcal C}\) be an arrangement of disks with a bijection between the disks and the clusters. Akitaya et al. [2] give an algorithm to test whether Open image in new window can be embedded onto \(\mathcal {D}_{\mathcal C}\) with the additional constraint that edges are routed through a set of pipes between the disks. Based on such an embedding, we prove that every clustered graph and every disk arrangement without pipe-disk intersections has a planar straight-line drawing where every vertex is embedded in the disk corresponding to its cluster. This result can be seen as an extension of the result by Alam et al. [3] who solely consider biconnected clusters. Moreover, we prove that it is \(\mathcal {NP} \)-hard to decide whether a clustered graph has such a straight-line drawing, if we permit pipe-disk intersections.

A point set $$\mathcal{S} \subseteq \mathbb {R}^2$$ is universal for a class $$\mathcal G$$ of planar graphs if every graph of $$\mathcal{G}$$ has a planar straight-line embedding on $$\mathcal{S}$$ . It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a subquadratic universal point set still remains one of the most fascinating open problems in Graph Drawing. In this paper we make a major step towards a solution for this problem. Motivated by the fact that each point set of size n in general position is universal for the class of n-vertex outerplanar graphs, we concentrate our attention on k-outerplanar graphs. We prove that they admit an $$O(n \log n)$$ -size universal point set in two distinct cases, namely when $$k=2$$ (2-outerplanar graphs) and when k is unbounded but each outerplanarity level is restricted to be a simple cycle (simply-nested graphs).

A graph is called a strong (resp. weak) bar 1-visibility graph if its vertices can be represented as horizontal segments (bars) in the plane so that its edges are all (resp. a subset of) the pairs of vertices whose bars have a $\epsilon$-thick vertical line connecting them that intersects at most one other bar. We explore the relation among weak (resp. strong) bar 1-visibility graphs and other nearly planar graph classes. In particular, we study their relation to 1-planar graphs, which have a drawing with at most one crossing per edge; quasi-planar graphs, which have a drawing with no three mutually crossing edges; the squares of planar 1-flow networks, which are upward digraphs with in- or out-degree at most one. Our main results are that 1-planar graphs and the (undirected) squares of planar 1-flow networks are weak bar 1-visibility graphs and that these are quasi-planar graphs.

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given \(k \ge 0\), a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For \(k=0\) and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of “layered” 1-bend drawings.

We describe a unified approach for studying book, point-set, and simultaneous embeddability problems of upward planar digraphs. The approach is based on a linear time strategy to compute an upward planar drawing of an upward planar digraph such that all vertices are collinear. Besides having impact in relevant application domains of graph drawing and computational geometry, the presented results open new research directions in the area of upward planarity with constraints of the positions of the vertices.

A point set $$S \subseteq \mathbb {R}^2$$ is universal for a class $$\mathcal G$$ if every graph of $$\mathcal{G}$$ has a planar straight-line embedding on S. It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a sub-quadratic universal point set for them is one of the most fascinating open problems in Graph Drawing. Motivated by the fact that outerplanarity is a key property for the existence of small universal point sets, we study 2-outerplanar graphs and provide for them a universal point set of size $$On \log n$$.

A directed path whose edges are assigned labels "up", "down", "right", or "left" is called four-directional, and three-directional if at most three out of the four labels are used. A direction-consistent embedding of an n-vertex four-directional path P on a set S of n points in the plane is a straight-line drawing of P where each vertex of P is mapped to a distinct point of S and every edge points to the direction specified by its label. We study planar direction-consistent embeddings of three- and four-directional paths and provide a complete picture of the problem for convex point sets.

Graph and cartographic visualization have the common objective to provide intuitive understanding of some underlying data. We consider a problem that combines aspects of both by studying the problem of fitting planar graphs on planar maps. After providing an NP-hardness result for the general decision problem, we identify sufficient conditions so that a fit is possible on a map with rectangular regions. We generalize our techniques to non-convex rectilinear polygons, where we also address the problem of efficient distribution of the vertices inside the map regions.

A point set P⊆ℝ2 is universal for a class $\cal G$ if every graph of ${\cal G}$ has a planar straight-line embedding into P. We prove that there exists a $O(n (\frac{\log n}{\log\log n})^2)$ size universal point set for the class of simply-nested n-vertex planar graphs. This is a step towards a full answer for the well-known open problem on the size of the smallest universal point sets for planar graphs [1, 5, 9].

In point-set-embeddability (PSE) problems one is given not just a graph that is to be drawn, but also a set of points in the plane that specify where the vertices of the graph can be placed. The problem class was introduced by Gritzmann et al. [3] twenty years ago. In their work and most other works on PSE problems, however, planarity of the output drawing was an essential requirement. Recent experiments on the readability of drawings [4] showed that polyline drawings with angles at edge crossings close to 90°. and a small number of bends per edge are just as readable as planar drawings. Motivated by these findings, Didimo et al. [2] recently introduced RAC drawings where pairs of crossing edges must form a right angle and, more generally, αAC drawings (for α∈ (0, 90°]) where the crossing angle must be at least α. As usual, edges may not overlap and may not go through vertices. We investigate the intersection of PSE and RAC/αAC.

We study the problem of Upward Point-Set Embeddability, that is the problem of deciding whether a given upward planar digraph $D$ has an upward planar embedding into a point set $S$. We show that any switch tree admits an upward planar straight-line embedding into any convex point set. For the class of $k$-switch trees, that is a generalization of switch trees (according to this definition a switch tree is a $1$-switch tree), we show that not every $k$-switch tree admits an upward planar straight-line embedding into any convex point set, for any $k \geq 2$. Finally we show that the problem of Upward Point-Set Embeddability is NP-complete.

Giordano, Liotta and Whitesides [1] developed an algorithm that, given an embedded planar st-digraph and a topological numbering ρ of its vertices, computes in O(n2) time a ρ-constrained upward topological book embedding with at most 2n–4 spine crossings per edge. The number of spine crossings per edge is asymptotically worst case optimal.

In this paper we study the problem of existence of a crossing-free acyclic hamiltonian path completion (for short, HP-completion) set for embedded upward planar digraphs. In the context of book embeddings, this question becomes: given an embedded upward planar digraph G, determine whether there exists an upward 2-page book embedding of G preserving the given planar embedding. Given an embedded st-digraph G which has a crossing-free HP-completion set, we show that there always exists a crossing-free HP-completion set with at most two edges per face of G. For an embedded N-free upward planar digraph G, we show that there always exists a crossing-free acyclic HP-completion set for G which, moreover, can be computed in linear time. For a width-k embedded planar st-digraph G, we show that it can be efficiently tested whether G admits a crossing-free acyclic HP-completion set.

Given an embedded planar acyclic digraph G , the acyclic hamiltonian path completion with crossing minimization (Acyclic- HPCCM) problem is to determine a hamiltonian path completion set of edges such that, when these edges are embedded on G , they create the smallest possible number of edge crossings and turn G to a hamiltonian acyclic digraph. In this paper, we present a linear time algorithm which solves the Acyclic-HPCCM problem on any outerplanar st -digraph G . The algorithm is based on properties of the optimal solution and an st-polygon decomposition of G . As a consequence of our result, we can obtain for the class of outerplanar st -digraphs upward topological 2-page book embeddings with minimum number of spine crossings.

An upward topological book embedding of a planar st -digraph G is an upward planar drawing of G such that its vertices are aligned along the vertical line, called the spine , and each edge is represented as a simple Jordan curve which is divided by the intersections with the spine (spine crossings ) into segments such that any two consecutive segments are located at opposite sides of the spine. When we treat the problem of obtaining an upward topological book embedding as an optimization problem, we are naturally interested in embeddings with the minimum possible number of spine crossing.