Goos Kant

Drawing planar graphs using the canonical ordering

AbstractWe introduce a new method to optimize the required area, minimum angle, and number of bends of planar graph drawings on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear-time-and-space algorithms can be designed for many graph-drawing problems. Our main results are as follows:Every triconnected planar graphG admits a planar convex grid drawing with straight lines on a (2n−4)×(n−2) grid, wheren is the number of vertices.Every triconnected planar graph with maximum degree 4 admits a planar orthogonal grid drawing on ann×n grid with at most [3n/2]+4 bends, and ifn>6, then every edge has at most two bends.Every planar graph with maximum degree 3 admits a planar orthogonal grid drawing with at most [n/2]+1 bends on an [n/2]×[n/2] grid.Every triconnected planar graphG admits a planar polyline grid drawing on a (2n−6)×(3n−9) grid with minimum angle larger than 2/d radians and at most 5n−15 bends, withd the maximum degree. These results give in some cases considerable improvements over previous results, and give new bounds in other cases. Several other results, e.g., concerning visibility representations, are included.

Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems

Abstract In this paper we extend the concept of the regular edge labeling for general plane graphs and for triconnected triangulated plane graphs to 4-connected triangulated plane graphs. We present two different linear time algorithms for constructing such a labeling. By using regular edge labeling, we present a new linear time algorithm for constructing rectangular dual of planar graphs. Our algorithm is simpler than previously known algorithms. The coordinates of the rectangular dual constructed by our algorithm are integers, while the one constructed by known algorithms are real numbers. Our second regular edge labeling algorithm is based on canonical ordering of 4-connected triangulated plane graphs. By using this technique, we present a new algorithm for constructing visibility representation of 4-connected planar graphs. Our algorithm reduces the size of the representation by a factor of 2 for such graphs.

On triangulating planar graphs under the four-connectivity constraint

Abstract. Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy 4-connectivity. A 4-connected planar graph has no separating triangles, i.e., cycles of length 3 which are not a face. We show that triangulating embedded planar graphs without introducing new separating triangles can be solved in linear time and space. If the initial graph had no separating triangle, the resulting triangulation is 4-connected. If the planar graph is not embedded, then deciding whether there exists an embedding with at most k separating triangles is NP-complete. For biconnected graphs a linear-time approximation which produces an embedding with at most twice the optimal number is presented. With this algorithm we can check in linear time whether a biconnected planar graph can be made 4-connected while maintaining planarity. Several related remarks and results are included.

2-Visibility Drawings of Planar Graphs

In a 2-visibility drawing the vertices of a given graph are represented by rectangular boxes and the adjacency relations are expressed by horizontal and vertical lines drawn between the boxes. In this paper we want to emphasize this model as a practical alternative to other representations of graphs, and to demonstrate the quality of the produced drawings. We give several approaches, heuristics as well as provably good algorithms, to represent planar graphs within this model. To this, we present a polynomial time algorithm to compute a bend-minimum orthogonal drawing under the restriction that the number of bends at each edge is at most 1.

Planar graph augmentation problems

In this paper we investigate the problem of adding a minimum number of edges to a planar graph in such a way that the resulting graph is biconnected and still planar. It is shown that this problem is NP-complete. We present an approximation algorithm for this planar biconnectivity augmentation problem that has performance ratio 3/2 and uses O(n2 log n) time. An O(n3) approximation algorithm with performance ratio 5/4 is presented to make a biconnected planar graph triconnected by adding edges without losing planarity.

Two Algorithms for Finding Rectangular Duals of Planar Graphs

We present two linear-time algorithms for computing a regular edge labeling of 4-connected planar triangular graphs. This labeling is used to compute in linear time a rectangular dual of this class of planar graphs. The two algorithms are based on totally different frameworks, and both are conceptually simpler than the previous known algorithm and are of independent interests. The first algorithm is based on edge contraction. The second algorithm is based on the canonical ordering. This ordering can also be used to compute more compact visibility representations for this class of planar graphs.

A Better Heuristic for Orthogonal Graph Drawings

An orthogonal drawing of a graph is an embedding in the plane such that all edges are drawn as sequences of horizontal and vertical segments. We present a linear time and space algorithm to draw any connected graph orthogonally on a grid of size n×n with at most 2n+2 bends. Each edge is bent at most twice.

Augmenting Outerplanar Graphs

In this paper, we show that for outerplanar graphsGthe problem of augmentingGby adding a minimum number of edges such that the augmented graphGÂ? is planar and bridge-connected, biconnected, or triconnected can be solved in linear time and space. It is also shown that augmenting a biconnected outerplanar graph to a maximal outerplanar graph while minimizing the maximum degree can be achieved in polynomial time. These augmentation problems arise in the area of drawing outerplanar graphs.

Hexagonal Grid Drawings

In this paper we present a linear algorithm to draw triconnected planar graphs of degree 3 planar on a linear-sized hexagonal grid such that in at most one edge are bends. This algorithm can be used to draw this class of graphs planar with straight lines on a n/2 × n/2 grid, improving the best known grid bounds by a factor 4. We also show how to draw planar graphs of degree at most 3 planar with straight lines such that the minimum angle is ≥ π/6, thereby answering a question of Formann et al.

Area requirement of visibility representations of trees

Trees are among the most common structures in computing and many algorithms for drawing trees have been developed in the last years. Such algorithms usually adopt different drawing conventions and attempt to solve several optimization problems. The aim of this paper is to study two different types of drawing conventions for trees, namely 1-strong visibility representation and 2-strong visibility representation. For both of them we investigate the problem of minimizing the area of the representation. The contribution of the paper is twofold: (i) we prove tight lower and upper bounds on the area of such representations; and (ii) we provide linear-time algorithms that construct representations with optimal area.