Anchored Drawings of Planar Graphs

Lecture Notes in Computer Science, 2023

A rectangular drawing of a planar graph G is a planar drawing of G in which vertices are mapped to grid points, edges are mapped to horizontal and vertical straight-line segments, and faces are drawn as rectangles. Sometimes this latter constraint is relaxed for the outer face. In this paper, we study rectangular drawings in which the edges have unit length. We show a complexity dichotomy for the problem of deciding the existence of a unit-length rectangular drawing, depending on whether the outer face must also be drawn as a rectangle or not. Specifically, we prove that the problem is NP-complete for biconnected graphs when the drawing of the outer face is not required to be a rectangle, even if the sought drawing must respect a given planar embedding, whereas it is polynomial-time solvable, both in the fixed and the variable embedding settings, if the outer face is required to be drawn as a rectangle.

Graph Algorithms and Applications 4, 2006

2011

The classical Fáry's theorem from the 1930s states that every planar graph can be drawn as a straight-line drawing. In this paper, we extend Fáry's theorem to non-planar graphs. More specifically, we study the problem of drawing 1-planar graphs with straight-line edges. A 1-planar graph is a sparse non-planar graph with at most one crossing per edge. We give a characterisation of those 1planar graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. We also show that there are 1-planar graphs for which every straight-line drawing has exponential area. To our best knowledge, this is the first result to extend Fáry's theorem to non-planar graphs.

Journal of Graph Algorithms and Applications, 2007

In this paper we consider the problem of drawing a planar graph using circular arcs as edges, given a one-to-one mapping between the vertices of the graph and a set of points in the plane. If for every edge we have only two possible circular arcs, then a simple reduction to 2SAT yields an O(n 2) algorithm to find out if a drawing with no crossings can be realized, where n is the number of vertices in the graph. We present an improved O(n 7/4 polylog n) time algorithm for this problem. For the special case where the possible circular arcs for each edge are of the same length, we present an even more efficient algorithm that runs in O(n 3/2 polylog n) time. We also consider two related optimization versions of the problem. First we show that minimizing the number of crossings is NP-hard. Second we show that maximizing the number of edges that can be realized without crossings is also NP-hard. Finally, we show that if we have three or more possible circular arcs per edge, deciding whether a drawing with no crossings can be realized is NP-hard.

Lecture Notes in Computer Science, 2010

A rectilinear drawing is an orthogonal grid drawing without bends, possibly with edge crossings, without any overlapping between edges, between vertices, or between edges and vertices. Rectilinear drawings without edge crossings (planar rectilinear drawings) have been extensively investigated in graph drawing. Testing rectilinear planarity of a graph is NP-complete . Restricted cases of the planar rectilinear drawing problem, sometimes called the "no-bend orthogonal drawing problem", have been well studied (see, for example, ).

Computational Geometry, 2005

Let C be the family of 2D curves described by concave functions, let G be a planar graph, and let L be a linear ordering of the vertices of G. L is a curve embedding of G if for any given curve Λ ∈ C there exists a planar drawing of G such that: (i) the vertices are constrained to be on Λ with the same ordering as in L, and (ii) the edges are polylines with at most one bend. Informally speaking, a curve embedding can be regarded as a two-page book embedding in which the spine is bent. Although deciding whether a graph has a two-page book embedding is an NP-hard problem, in this paper it is proven that every planar graph has a curve embedding which can be computed in linear time. Applications of the concept of curve embedding to upward drawability and point-set embeddability problems are also presented.

2015

We consider the problem of finding a planar embedding of a graph at fixed vertex locations that minimizes the total edge length. The problem is known to be NP-hard. We give polynomial time algorithms achieving an O(n log n) approximation for paths and matchings, and an O(n) approximation for general graphs.

Lecture Notes in Computer Science, 2011

We study the complexity of the problem of finding nonplanar rectilinear drawings of graphs. This problem is known to be NPcomplete. We consider natural restrictions of this problem where constraints are placed on the possible orientations of edges. In particular, we show that if each edge has prescribed direction "left", "right", "down" or "up", the problem of finding a rectilinear drawing is polynomial, while finding such a drawing with the minimum area is NP-complete. When assigned directions are "horizontal" or "vertical" or a cyclic order of the edges at each vertex is specified, the problem is NP-complete. We show that these two NP-complete cases are fixed parameter tractable in the number of vertices of degree 3 or 4.

Computational Geometry, 1998

Given a plane graph G, we wish to draw it in the plane in such a way that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints. An additional objective is to minimize the size of the resulting grid. It is known that each plane graph can be drawn in such a way in an (n-2) x (n-2) grid (for n ~> 3), and that no grid smaller than (2n/3-1) x (2n/3-1) can be used for this purpose, if n is a multiple of 3. In fact, for all n ~> 3, each dimension of the resulting grid needs to be at least [2(n-1)/3J, even if the other one is allowed to be unbounded. In this paper we show that this bound is fight by presenting a grid drawing algorithm that produces drawings of width 12(n-1)/33. The height of the produced drawings is bounded by 4/2(n-1)/3J-1. Our algorithm runs in linear time and is easy to implement.

Journal of Graph Algorithms and Applications, 2018

We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw two collinear edges of the same vertex). Let n denote the number of vertices of a graph. We show that trees can be drawn with 3n/4 straight-line segments on a polynomial grid, and with n/2 straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with (8n -17)/3 segments on an O(n) × O(n 2 ) grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n/2 edges on an O(n) × O(n 2 ) grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n -11)/3 arcs. This provides a significant improvement over the lower bound of 2n for line segments for a nontrivial graph class.