This paper appears as Technical Report 95-07, Department of Computer Science, University of Newcastle, Newcastle NSW 2308 Australia.

Graphs are ubiquitous, finding applications in domains ranging from software engineering to computational biology. While graph theory and graph algorithms are some of the oldest, most studied fields in computer science, the problem of visualizing graphs is comparatively young. This problem, known as graph drawing, is that of transforming combinatorial graphs into geometric drawings for the purpose of visualization. Most published algorithms for drawing general graphs model the drawing problem with a physical analogy, representing a graph as a system of springs and other physical elements and then simulating the relaxation of this physical system. Solving the graph drawing problem involves both choosing a physical model and then using numerical optimization to simulate the physical system. In this

An orthogonal drawing of a graph is a drawing such that nodes are placed on grid points and edges are drawn as sequences of vertical and horizontal segments. In this paper we present linear time algorithms that produce orthogonal drawings of graphs with n nodes. If the maximum degree is four, then the drawing produced by our first algorithm needs area at most (roughly) 0:76n 2 , and introduces at most 2n + 2 bends. Also, each edge of such a drawing has at most two bends. Our algorithm is based on forming and placing pairs of vertices of the graph. If the maximum degree is three, then the drawing produced by our second algorithm needs at most (roughly) 1 4 n 2 area and, if the graph is biconnected, at most b n 2 c + 3 bends. These upper bounds match the upper bounds known for planar graphs of maximum degree 3. This algorithm produces optimal drawings (within a constant of 2) with respect to the number of bends, since there is a lower bound of n 2 + 1 in the number of bends fo...

Given an orthogonal representation H with n vertices and bends, we study the problem of computing a planar orthogonal drawing of H with small area. This problem has direct applications to the development of practical graph drawing techniques for information visualization and VLSI layout. In this paper, we introduce the concept of turn-regularity of an orthogonal representation H, provide combinatorial characterizations of it, and show that if H is turn-regular (i.e., all its faces are turn-regular), then a planar orthogonal drawing of H with minimum area can be computed in O(n) time, and a planar orthogonal drawing of H with minimum area and minimum total edge length within that area can be computed in O(n 7/4 log n) time. We also apply our theoretical results to the design and implementation of new practical heuristic methods for constructing planar orthogonal drawings. An experimental study conducted on a test suite of orthogonal representations of randomly generated biconnected 4-planar graphs shows that the percentage of turn-regular faces is quite high and that our heuristic drawing methods perform better than previous ones.

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 5 2n 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.

) Isabel F. Cruz 1 and Ashim Garg 2 1 Department of Electrical Engineering and Computer Science Tufts University Medford, MA 02155, USA 2 Department of Computer Science Brown University Providence, RI 02912--1910, USA Abstract. Constraint-based graph drawing systems provide expressive power and flexibility. Previously proposed approaches make use of general constraint solvers, which are inefficient, and of textual specification of constraints, which can be long and difficult to understand. In this paper we propose the use of a constraint-based visual language for constructing planar drawings of trees, series-parallel graphs, and acyclic digraphs in linear time. A graph drawing system based on our approach can therefore provide the power of constraint-based graph drawing, the simplicity of visual specifications, and the computational efficiency that is typical of the algorithmic-based approaches. 1 Introduction It is common practice to explain the layout of a graph using pictu...

A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a non-algorithmic, graph-theoretic characterization of those directed graphs with a single source that admit an upward drawing. We present an efficient algorithm to test whether a given single-source acyclic digraph has a plane upward drawing and, if so, to find a representation of one such drawing. The algorithm decomposes the graph into biconnected and triconnected components, and defines conditions for merging the components into an upward drawing of the original graph. For the triconnected components we provide a linear algorithm to test whether a given plane representation admits an upward drawing with the same faces and outer face, which also gives a simpler (and algorithmic) proof of Thomassen's result. The entire testing algorithm (for general single source directed acyclic graphs) operates in O(n²) time and...

In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the D-dimensional (D >= 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane.

Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NP-hard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQR-trees, which is able to find a crossing minimum solution.

We study straight-line drawings of 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 5n/2 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). Drawings of non-planar graphs with few slopes are also considered. For example, interval graphs, co-comparability graphs and AT-free graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree.