We prove the NP-completeness of the following problem. Given a set $S$ of $n$ slopes and an integer $k\geq 1$, is it possible to draw a complete graph on $k$ vertices in the plane using only slopes from $S$? Equivalently, does there exist a set $K$ of $k$ points in general position such that the slope of every segment between two points of $K$ is in $S$? We then present a polynomial algorithm for this question when $n\leq 2k-c$, conditional on a conjecture of R.E. Jamison. For $n=k$, an algorithm in $\mathcal{O}(n^4)$ was proposed by Wade and Chu. For this case, our algorithm is linear and does not rely on Jamison's conjecture.
We consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that $\lceil\fracΔ{2}\rceil$ edge slopes suffice for outerplanar drawings of outerplanar graphs with maximum degree $Δ\geq 3$. This matches the obvious lower bound. We also show that $\lceil\fracΔ{2}\rceil+1$ edge slopes suffice for drawings of general graphs, improving on the previous bound of $Δ+1$. Furthermore, we improve previous upper bounds on the number of slopes needed for planar dra…
The slope number of a graph $G$ is the smallest number of slopes needed for the segments representing the edges in any straight-line drawing of $G$. It serves as a measure of the visual complexity of a graph drawing. Several bounds on the slope number for particular graph classes have been established, both in the planar and the non-planar setting. Moreover, the slope number can also be defined for directed graphs and upward planar drawings. We study upward planar straight-line drawings that use only a constant number of slopes. In particular, for a fixed number $k$ of slopes, we are intereste…
We consider straight-line outerplanar drawings of outerplanar graphs in which a small number of distinct edge slopes are used, that is, the segments representing edges are parallel to a small number of directions. We prove that $Δ-1$ edge slopes suffice for every outerplanar graph with maximum degree $Δ\ge 4$. This improves on the previous bound of $O(Δ^5)$, which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every $Δ\ge 4$ there is an outerplanar graph with maximum degree $Δ$ that requires at least $Δ-1$ distinct edge slopes in an outerplana…
We prove that every set $\mathcal S$ of $Δ$ slopes containing the horizontal slope is universal for $1$-bend upward planar drawings of bitonic $st$-graphs with maximum vertex degree $Δ$, i.e., every such digraph admits a $1$-bend upward planar drawing whose edge segments use only slopes in $\mathcal S$. This result is worst-case optimal in terms of the number of slopes, and, for a suitable choice of $\mathcal S$, it gives rise to drawings with worst-case optimal angular resolution. In addition, we prove that every such set $\mathcal S$ can be used to construct $2$-bend upward planar drawings o…
We describe a set of $Δ-1$ slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree $Δ\geq 4$; this establishes a new upper bound of $Δ-1$ on the 1-bend planar slope number. By universal we mean that every planar graph of degree $Δ$ has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was $\frac{3}{2} (Δ-1)$ (the known lower bound being $\frac…
We show that the 1-planar slope number of 3-connected cubic 1-planar graphs is at most 4 when edges are drawn as polygonal curves with at most 1 bend each. This bound is obtained by drawings whose vertex and crossing resolution is at least $π/4$. On the other hand, if the embedding is fixed, then there is a 3-connected cubic 1-planar graph that needs 3 slopes when drawn with at most 1 bend per edge. We also show that 2 slopes always suffice for 1-planar drawings of subcubic 1-planar graphs with at most 2 bends per edge. This bound is obtained with vertex resolution $π/2$ and the drawing is RAC…
We introduce and study level-planar straight-line drawings with a fixed number $λ$ 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(λ n^…
In an upward planar 2-slope drawing of a digraph, edges are drawn as straight-line segments in the upward direction without crossings using only two different slopes. We investigate whether a given upward planar digraph admits such a drawing and, if so, how to construct it. For the fixed embedding scenario, we give a simple characterisation and a linear-time construction by adopting algorithms from orthogonal drawings. For the variable embedding scenario, we describe a linear-time algorithm for single-source digraphs, a quartic-time algorithm for series-parallel digraphs, and a fixed-parameter…
It is proved that every series-parallel digraph whose maximum vertex-degree is $Δ$ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of $Δ$ distinct slopes. This is shown to be worst-case optimal in terms of the number of slopes. Furthermore, our construction gives rise to drawings with optimal angular resolution $\fracπΔ$. A variant of the proof technique is used to show that (non-directed) reduced series-parallel graphs and flat series-parallel graphs have a (non-upward) one-bend planar drawing with $\lceil\fracΔ{2}\rceil$ distinct slopes if b…
This paper has been withdrawn by the authors. It has been replaced by the papers: "Drawings of Planar Graphs with Few Slopes and Segments" (math/0606450) and "Graph Drawings with Few Slopes" (math/0606446).
We introduce L-drawings, a novel paradigm for representing directed graphs aiming at combining the readability features of orthogonal drawings with the expressive power of matrix representations. In an L-drawing, vertices have exclusive $x$- and $y$-coordinates and edges consist of two segments, one exiting the source vertically and one entering the destination horizontally. We study the problem of computing L-drawings using minimum ink. We prove its NP-completeness and provide a heuristics based on a polynomial-time algorithm that adds a vertex to a drawing using the minimum additional ink. W…
The splitting number of a graph $G=(V,E)$ is the minimum number of vertex splits required to turn $G$ into a planar graph, where a vertex split removes a vertex $v \in V$, introduces two new vertices $v_1, v_2$, and distributes the edges formerly incident to $v$ among its two split copies $v_1, v_2$. The splitting number problem is known to be NP-complete. In this paper we shift focus to the splitting number of graph drawings in $\mathbb R^2$, where the new vertices resulting from vertex splits can be re-embedded into the existing drawing of the remaining graph. We first provide a non-uniform …
We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opp…
Computing a minimum-area planar straight-line drawing of a graph is known to be NP-hard for planar graphs, even when restricted to outerplanar graphs. However, the complexity question is open for trees. Only a few hardness results are known for straight-line drawings of trees under various restrictions such as edge length or slope constraints. On the other hand, there exist polynomial-time algorithms for computing minimum-width (resp., minimum-height) upward drawings of trees, where the height (resp., width) is unbounded. In this paper we take a major step in understanding the complexity of th…
An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at 45 degrees line-segments. For such drawings to be readable, special care is needed in order to keep the number of bends small. As the problem of finding planar octilinear drawings of minimum number of bends is NP-hard, in this paper we focus on upper and lower bounds. From a recent result of Keszegh et al. on the slope number of planar graphs, we can derive an upper bound of 4n-10 bends for 8-planar graphs with n vertices. We considerably improve this general bound a…
Algorithmic extension problems of partial graph representations such as planar graph drawings or geometric intersection representations are of growing interest in topological graph theory and graph drawing. In such an extension problem, we are given a tuple $(G,H,\mathcal{H})$ consisting of a graph $G$, a connected subgraph $H$ of $G$ and a drawing $\mathcal{H}$ of $H$, and the task is to extend $\mathcal{H}$ into a drawing of $G$ while maintaining some desired property of the drawing, such as planarity. In this paper we study the problem of extending partial 1-planar drawings, which are drawi…
A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which i…
The \emph{segment number} of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively. In this paper, we study three variants of the segment number: for planar graphs, we consider crossing-free polyline drawings in 2D; for arbitrary graphs, we consider crossing-free straight-line drawings in 3D and straight-line drawings with crossings in 2D. We first construct an infinite family of planar graphs wher…
A \emph{Stick graph} is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a `ground line,' a line with slope $-1$. It is an open question to decide in polynomial time whether a given bipartite graph $G$ with bipartition $A\cup B$ has a Stick representation where the vertices in $A$ and $B$ correspond to horizontal and vertical segments, respectively. We prove that $G$ has a Stick representation if and only if there are orderings of $A$ and $B$ such that $G$'s bipartite adjacency matr…
