Right-Angle Crossings Research Articles

Fixed-Parameter Algorithms for Computing Bend-Restricted RAC Drawings of Graphs

In a right-angle crossing (RAC) drawing of a graph, each edge is represented as a polyline and edge crossings must occur at an angle of exactly $90^\circ$, where the number of bends on such polylines is typically restricted in some way. While structural and topological properties of RAC drawings have been the focus of extensive research and particular attention has been paid to RAC drawings with at most $0$, $1$, $2$ and $3$ bends per edge, little was known about the boundaries of tractability for computing such drawings. In this paper, we initiate the study of bend-restricted RAC drawings from the viewpoint of parameterized complexity. In particular, we establish that computing a RAC drawing of an input graph $G$ with at most $b$ bends where each edge $e$ has a prescribed upper-bound $0\leq \beta(e) \leq 3$ on the number of bends it can support (or determining that none exists) is: fixed-parameter tractable parameterized by the feedback edge number of $G$, and fixed-parameter tractable parameterized by $b$ plus the vertex cover number of $G$.

Compact drawings of 1-planar graphs with right-angle crossings and few bends

We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two pairs of crossing edges share two vertices.We study the relations of these beyond-planar graph classes (beyond-planar graphs is a collective term for the primary attempts to generalize the planar graphs) to right-angle crossing (RAC) graphs that admit compact drawings on the grid with few bends. We present four drawing algorithms that preserve the given embeddings. First, we show that every n-vertex NIC-planar graph admits a NIC-planar RAC drawing with at most one bend per edge on a grid of size O(n)×O(n). Then, we show that every n-vertex 1-planar graph admits a 1-planar RAC drawing with at most two bends per edge on a grid of size O(n3)×O(n3). Finally, we make two known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at most one bend per edge and for drawing IC-planar RAC graphs straight-line.

On RAC drawings of 1-planar graphs

A drawing of a graph is 1-planar if each edge is crossed at most once. A graph is 1-planar if it has a 1-planar drawing. A k-bend RAC (Right Angle Crossing) drawing of a graph is a polyline drawing where each edge has at most k bends and edges cross only at right angles. A graph is k-bend RAC if it has a k-bend RAC drawing. A 0-bend RAC graph (drawing) is also called a straight-line RAC graph (drawing). The relationships between 1-planar and k-bend RAC graphs have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC and straight-line RAC graphs that are not 1-planar. The existence of 1-planar straight-line RAC drawings has been proven only for restricted families of 1-planar graphs. Two of the main questions still open are: (i) What is the complexity of deciding whether a graph has a drawing that is both 1-planar and straight-line RAC? (ii) Does every 1-planar graph have a drawing that is both 1-planar and 1-bend RAC? In this paper we answer these two questions. Namely, we prove an NP-hardness result for the first question, and we positively answer the second question by describing a drawing algorithm for 1-planar graphs.

Simultaneous visibility representations of plane st-graphs using L-shapes

Let 〈Gr,Gb〉 be a pair of plane st-graphs with the same vertex set V. A simultaneous visibility representation with L-shapes of 〈Gr,Gb〉 is a pair of bar visibility representations 〈Γr,Γb〉 such that, for every vertex v∈V, Γr(v) and Γb(v) are a horizontal and a vertical segment, respectively, which share an end-point. In other words, every vertex is drawn as an L-shape, every edge of Gr is a vertical visibility segment, and every edge of Gb is a horizontal visibility segment. Also, no two L-shapes intersect each other. An L-shape has four possible rotations, and we assume that each vertex is given a rotation for its L-shape as part of the input. Our main results are: (i) a characterization of those pairs of plane st-graphs admitting such a representation, (ii) a quadratic time algorithm to recognize them, and (iii) a linear time drawing algorithm if the test is positive. As an application, starting from a simultaneous visibility representation with L-shapes, we show how to compute a simultaneous embedding of the two graphs with at most two bends per edge and right-angle crossings.

Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends

Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is a drawing of the two graphs where each graph is drawn planar, no two edges overlap, and edges of one graph can cross edges of the other graph only at right angles. In the geometric version of the problem, vertices are drawn as points and edges as straight-line segments. It is known, however, that even pairs of very simple classes of planar graphs (such as wheels and matchings) do not always admit a geometric RAC simultaneous drawing. In order to enlarge the class of graphs that admit RAC simultaneous drawings, we allow edges to have bends. We prove that any pair of planar graphs admits a RAC simultaneous drawing with at most six bends per edge. For more restricted classes of planar graphs (e.g., matchings, paths, cycles, outerplanar graphs, and subhamiltonian graphs), we significantly reduce the required number of bends per edge. All our drawings use quadratic area.

EVERY OUTER-1-PLANE GRAPH HAS A RIGHT ANGLE CROSSING DRAWING

There is strong empirical evidence that human perception of a graph drawing is negatively correlated with the number of edge crossings. However, recent experiments show that one can reduce the negative effect by ensuring that the edges that cross do so at large angles. These experiments have motivated a number of mathematical and algorithmic studies of “right angle crossing (RAC)” drawings of graphs, where the edges cross each other perpendicularly. In this paper we give an algorithm for constructing RAC drawings of “outer-1-plane” graphs, that is, topological graphs in which each vertex appears on the outer face, and each edge crosses at most one other edge. The drawing algorithm preserves the embedding of the input graph. This is one of the few algorithms available to construct RAC drawings.

Graphs that admit right angle crossing drawings

We consider right angle crossing ( RAC) drawings of graphs in which the edges are represented by polygonal arcs and any two edges can cross only at a right angle. We show that if a graph with n vertices admits a RAC drawing with at most 1 bend or 2 bends per edge, then the number of edges is at most 6.5 n and 74.2 n, respectively. This is a strengthening of a recent result of Didimo et al.

Drawing graphs with right angle crossings

Cognitive experiments show that humans can read graph drawings in which all edge crossings are at right angles equally well as they can read planar drawings; they also show that the readability of a drawing is heavily affected by the number of bends along the edges. A graph visualization whose edges can only cross perpendicularly is called a RAC (Right Angle Crossing) drawing. This paper initiates the study of combinatorial and algorithmic questions related to the problem of computing RAC drawings with few bends per edge. Namely, we study the interplay between number of bends per edge and total number of edges in RAC drawings. We establish upper and lower bounds on these quantities by considering two classical graph drawing scenarios: The one where the algorithm can choose the combinatorial embedding of the input graph and the one where this embedding is fixed.

Notes on large angle crossing graphs

A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges in G intersect at an angle of at least a. The concept of right angle crossing (RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown that any RAC graph with n vertices has at most 4n-10 edges and that there are infinitely many values of n for which there exists a RAC graph with n vertices and 4n-10 edges. In this paper, we give upper and lower bounds for the number of edges in aAC graphs for all 0 < a < Pi/2.