Drawings of planar graphs with few slopes and segments

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.     

Rectangular drawings of plane graphs without designated corners

This paper addresses the problem of finding rectangular drawings of plane graphs, in which each vertex is drawn as a point, each edge is drawn as a horizontal or a vertical line segment, and the contour of each face is drawn as a rectangle. A graph is a 2–3 plane graph if it is a plane graph and each vertex has degree 3 except the vertices on the outer face which have degree 2 or 3. A necessary and sufficient condition for the existence of a rectangular drawing has been known only for the case where exactly four vertices of degree 2 on the outer face are designated as corners in a 2–3 plane graph G. In this paper we establish a necessary and sufficient condition for the existence of a rectangular drawing of G for the general case in which no vertices are designated as corners. We also give a linear-time algorithm to find a rectangular drawing of G if it exists.     

Bold graph drawings

When a graph is drawn in a classical manner, its vertices are shown as small disks and its edges with a positive width; zero-width edges and zero-size vertices exist only in theory. Let r denote the radius of the disks that show vertices and w the width of edges. We give a list of conditions that make such a drawing good and that apply to not necessarily planar graphs. We show that if r<w, a vertex must have constant degree for a drawing to satisfy the conditions, and if r?w, a vertex can have any degree. We also give an algorithm that, for a given drawing and values for r and w, determines whether the bold drawing satisfies the conditions. Furthermore, we show how to maximize r and/or w without violating the conditions in polynomial time.     

Voronoi drawings of trees

We study the problem of characterizing sets of points whose Voronoi diagrams are trees and if so, what are the combinatorial properties of these trees. The second part of the problem can be naturally turned into the following graph drawing question: Given a tree T, can one represent T so that the resulting drawing is a Voronoi diagram of some set of points? We investigate the problem both in the Euclidean and in the Manhattan metric. The major contributions of this paper are as follows.

• We characterize those trees that can be drawn as Voronoi diagrams in the Euclidean metric.

• We characterize those sets of points whose Voronoi diagrams are trees in the Manhattan metric.

• We show that the maximum vertex degree of any tree that can be drawn as a Manhattan Voronoi diagram is at most five and prove that this bound is tight.

• We characterize those binary trees that can be drawn as Manhattan Voronoi diagrams.

Crossing-constrained hierarchical drawings

We study the problem of computing hierarchical drawings of layered graphs when some pairs of edges are not allowed to cross. We show that deciding the existence of a drawing satisfying at least k non-crossing constraints from a given set is NP-hard, even if the graph is 2-layered and even when the permutation of the vertices on one side of the bipartition is fixed. We then propose simple constant-ratio approximation algorithms for the optimization version of the problem, which requires to find a maximum realizable subset of constraints, and we discuss how to extend the well-known hierarchical approach for creating layered drawings of directed graphs so as to minimize the number of edge crossings while maximizing the number of satisfied constraints.     

Area-efficient algorithms for straight-line tree drawings

We investigate several straight-line drawing problems for bounded-degree trees in the integer grid without edge crossings under various types of drawings: (1) upward drawings whose edges are drawn as vertically monotone chains, a sequence of line segments, from a parent to its children, (2) order-preserving drawings which preserve the left-to-right order of the children of each vertex, and (3) orthogonal straight-line drawings in which each edge is represented as a single vertical or horizontal segment.

Main contribution of this paper is a unified framework to reduce the upper bound on area for the straight-line drawing problems from O(nlogn) (Crescenzi et al., 1992) to O(nloglogn). This is the first solution of an open problem stated by Garg et al. (1993). We also show that any binary tree admits a small area drawing satisfying any given aspect ratio in the orthogonal straight-line drawing type.

Our results are briefly summarized as follows. Let T be a bounded-degree tree with n vertices. Firstly, we show that T admits an upward straight-line drawing with area O(nloglogn). If T is binary, we can obtain an O(nloglogn)-area upward orthogonal drawing in which each edge is drawn as a chain of at most two orthogonal segments and which has O(n/logn) bends in total. Secondly, we present O(nloglogn)-area (respectively, -volume) orthogonal straight-line drawing algorithms for binary trees with arbitrary aspect ratios in 2-dimension (respectively, 3-dimension). Finally, we present some experimental results which shows the area requirements, in practice, for (order-preserving) upward drawing are much smaller than theoretical bounds obtained through analysis.     

An algorithm for constructing star-shaped drawings of plane graphs

A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing.In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.     

Linkless symmetric drawings of series parallel digraphs

In this paper, we present a linear time algorithm for constructing linkless drawings of series parallel digraphs with maximum number of symmetries. Linkless drawing in three dimensions is a natural extension to planar drawing in two dimensions. Symmetry is one of the most important aesthetic criteria in graph drawing. More specifically, we present two algorithms: a symmetry finding algorithm which finds maximum number of three dimensional symmetries, and a drawing algorithm which constructs linkless symmetric drawings of series parallel digraphs in three dimensions.     

Graph parameters measuring neighbourhoods in graphs—Bounds and applications

The neighbourhood-width of a graph G=(V,E) is introduced in [F. Gurski, Linear layouts measuring neighbourhoods in graphs, Discrete Math. 306 (15) (2006) 1637-1650.] as the smallest integer k such that there is a linear layout ?:V→{1,…,|V|} such that for every 1?i<|V| the vertices u with ?(u)?i can be divided into at most k subsets each members having the same neighbours with respect to the vertices v with ?(v)>i.In this paper we show first bounds for the neighbourhood-width of general graphs, caterpillars, trees and grid graphs and give applications of the layout parameter neighbourhood-width in graph drawing and VLSI design.     

Radial drawings of graphs: Geometric constraints and trade-offs

This paper studies how to compute radially layered drawings of graphs by taking into account additional geometric constraints which correspond to typical aesthetic and semantic requirements for the visualization. The following requirements are considered: vertex centrality, edge crossings, curve complexity, and radial distribution of the vertices. Trade-offs among these requirements are discussed and different linear-time drawing algorithms are presented.     

Graph layering by promotion of nodes

This work contributes to the wide research area of visualization of hierarchical graphs. We present a new polynomial-time heuristic which can be integrated into the Sugiyama method for drawing hierarchical graphs. Our heuristic, which we call Promote Layering (PL), is applied to the output of the layering phase of the Sugiyama method. PL is a simple and easy to implement algorithm which decreases the number of so-called dummy (or virtual) nodes in a layered directed acyclic graph. In particular, we propose applying PL after the longest-path layering algorithm and we present an extensive empirical evaluation of this layering technique.     

Crooked diagrams with few slopes

A natural and practical criterion in the preparation of diagrams of ordered sets is to minimize the number of different slopes used for the edges. For any diagram this is at least the maximum number of upper covers and of lower covers of any element. While this maximum degree is not always enough we show that it is as long as any edge joining a covering pair may be bent, to produce a crooked diagram.     

A framework and algorithms for circular drawings of graphs

In this paper, we present a framework and two linear time algorithms for obtaining circular drawings of graphs. The first technique produces circular drawings of biconnected graphs and finds a zero crossing circular drawing if one exists. The second technique finds multiple embedding circle drawings. Techniques for the reduction of edge crossings are also discussed. Results of experimental studies are included.     

Area requirement of graph drawings with few crossings per edge

In this paper we study how to compute compact straight-line drawings of planar graphs with a limited number of crossings per edge. We prove that every outerplanar graph can be drawn in $O(n\log n)$ area using a sub-linear number of crossings per edge, and that for any given number $\epsilon >0$, every outerplanar graph admits an $O(n^{1+\epsilon })$ area drawing with $O(n^{1?\epsilon })$ crossing per edge. The drawing algorithms run in linear time and can be also generalized to another meaningful sub-family of series–parallel graphs with bounded vertex-degree.     

Graph pegging numbers

In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like “pegging moves” allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging number (respectively, the optimal pegging number) of a graph is the minimum number of pegs such that for every (respectively, some) distribution of that many pegs on the graph, any vertex can be reached by a sequence of pegging moves. We prove several basic properties of pegging and analyze the pegging number and optimal pegging number of several classes of graphs, including paths, cycles, products with complete graphs, hypercubes, and graphs of small diameter.     

Planar configurations which determine few slopes

P. R. Scott conjectured, and Peter Ungar recently proved, that n noncollinear points in the plane determine at least n–1 slopes. This paper describes and characterizes the four known infinite classes of configurations where this minimum is achieved.Supported in part by NSF Grant MCS-80-02543 and NSF Epscor Grant IPS-80-11451     

The structure of sets with few sums along a graph

We present common generalizations of some structure results of Freiman, Ruzsa, Balog-Szemerédi and Laczkovich-Ruzsa. We also give some applications to Combinatorial Geometry and Algebra, some of which generalize the aforementioned structure results even further.     

Crossing minimization in extended level drawings of graphs

The most popular method of drawing directed graphs is to place vertices on a set of horizontal or concentric levels, known as level drawings. Level drawings are well studied in Graph Drawing due to their strong application for the visualization of hierarchy in graphs. There are two drawing conventions: Horizontal drawings use a set of parallel lines and radial drawings use a set of concentric circles.In level drawings, edges are only allowed between vertices on different levels. However, many real world graphs exhibit hierarchies with edges between vertices on the same level. In this paper, we initiate the new problem of extended level drawings of graphs, which was addressed as one of the open problems in social network visualization, in particular, displaying centrality values of actors. More specifically, we study minimizing the number of edge crossings in extended level drawings of graphs. The main problem can be formulated as the extended one-sided crossing minimization problem between two adjacent levels, as it is folklore with the one-sided crossing minimization problem in horizontal drawings.We first show that the extended one-sided crossing minimization problem is NP-hard for both horizontal and radial drawings, and then present efficient heuristics for minimizing edge crossings in extended level drawings. Our extensive experimental results show that our new methods reduce up to 30% of edge crossings.