Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows: ffl We devise a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing, and we show its applications. ffl We present several efficient dynamic drawing algorithms for trees, series-parallel digraphs, planar st-digraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straight-line, polyline, visibility), and update the drawing in a smooth way.

Abstract. Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar st-graphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(log n) running time using n = log n processors in the EREW PRAM model, n being the number of vertices. Key words. parallel algorithms, parallel computation, graph algorithms, planar st-graphs, transitive closure, reachability, planar point location, computational geometry, fractional cascading, graph drawing, visibility AMS(MOS) subject classi cations. 68E05, 68C05, 68C25 1. Introduction. Planar st-graphs

We study the problem of drawing a graph in the plane so that the vertices of the graph are rectangles that are aligned with the axes, and the edges of the graph are horizontal or vertical lines-of-sight. Such a drawing

This paper studies 3-dimensional visibility representations of graphs in which objects in 3-d correspond to vertices and vertical visibilities between these objects correspond to edges. We ask which classes of simple objects are universal, i.e. powerful enough to represent all graphs. In particular, we show that there is no constant k for which the class of all polygons having k or fewer sides is universal. However, we show by construction that every graph on n vertices can be represented by polygons each having at most 2n sides. The construction can be carried out by an O(n ) algorithm. We also study the universality of classes of simple objects (translates of a single, not necessarily polygonal object) relative to cliques Kn and similarly relative to complete bipartite graphs Kn;m .

In this paper we present an extensive experimental study...

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.

A k-path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing k-path queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes k-path queries in output-sensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.

We study the area requirement of bar-visibility and rectangle-visibility representations of trees in the plane. We prove asymptotically tight lower and upper bounds on the area of such representations, and give linear-time algorithms that construct representations with asymptotically optimal area.

Let S be a collection of n non-intersecting line segments in the plane in general position. Two segments u and v are defined as visible if a line segment could be drawn from some point of u to some point of v that intersects no other segment in S. The full visibility graph associated with S is denoted as G(S), and defined to be the graph whose n vertices correspond to the line segments of S and whose edge set represents the visibility relation between pairs of segments. A worst case optimal O(n 2 ) time and space algorithm for computing G(S) is presented.

Abstract. A triangulated polygon is a 2-connected maximal outerplanar graph. A unit bar-visibility graph (UBVG for short) is a graph whose vertices can be represented by disjoint, horizontal, unit-length bars in the plane so that two vertices are adjacent if and only if there is a nondegenerate, unobstructed, vertical band of visibility between the corresponding bars. We give combinatorial and geometric characterizations of the triangulated polygons that are UBVGs. To each triangulated polygon G we assign a character string with the property that G is a UBVG if and only if the string satisfies a certain regular expression. Given a string that satisfies this condition, we describe a linear-time algorithm that uses it to produce a UBV layout of G. 1