Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upward planarity testing. We present several characterizations of upward planarity and describe upward planarity testing algorithms for special classes of digraphs, such as embedded digraphs and single-source digraphs. We also sketch the proof of NP-completeness of upward planarity testing.

We give an efficient algorithm for maintaining a minimum spanning forest of a plane graph subject to on-line modifications. The modifications supported include changes in the edge weights, and insertion and deletion of edges and vertices which are consistent with the given embedding. Our algorithm runs in O(log n) time per operation and O(n) space.

We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs 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. -- Every triconnected planar graph G can be drawn convexly with straight lines on an (2n \Gamma 4) \Theta (n \Gamma 2) grid, where n is the number of vertices. -- Every triconnected planar graph with maximum degree four can be drawn orthogonally on an n \Theta n grid with at most d 3n 2 e + 4, and if n ? 6 then every edge has at most two bends. -- Every 3-planar graph G can be drawn with at most b n 2 c + 1 bends on an b n 2 c \Theta b n 2 c grid. -- Every triconnected planar graph G can be drawn planar on an (2n \Gamma 6) \Theta (3n \Gamma 9) grid with minimum angle larger than 2 d radians and at most 5n \Gamma 15 bends, with d the maximum d...

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.

We discuss space-efficient encoding schemes for planar graphs and maps. Our results improve on the constants of previous schemes and can be achieved with simple encoding algorithms. They are near-optimal in number of bits per edge. 1 Introduction In this paper we discuss space-efficient binary encoding schemes for several classes of unlabeled connected planar graphs and maps. In encoding a graph we must encode the incidences among vertexes and edges. By maps we understand topological equivalence classes of planar embeddings of planar graphs. In encoding a map we are required to encode the topology of the embedding i.e., incidences among faces, edges, and vertexes, as well as the graph. Each map is an embedding of a unique graph, but a given graph may have multiple embeddings. Hence maps must require more bits to encode than graphs in some average sense. There are a number of recent results on space-efficient encoding. A standard adjacency list encoding of an unlabeled graph G requires...

The authors of this article believe there is or should be a research area appropriately referred to as computational topology. Its agenda includes the identification and formalization of topological questions in computer applications and the study of algorithms for topological problems. It is hoped this article can contribute to the creation of a computational branch of topology with a unifying influence on computing and computer applications. Keywords. Survey; topology, geometry, algorithms, computer applications. INTRODUCTION The title of this article combines computation with topology, suggesting a general research activity that studies the computational aspects of problems with topological flavor. What we have in mind is distinctly different from studying the topology of computing or the computer animation of topology. Computational studies of topological questions can be found in the mathematics and the computer science literature, but no concerted effort is apparent. The auth...

We present a linear-time algorithm that, given an n-vertex planar graph G, finds an embedding of G into a (2n \Gamma 4) \Theta (n \Gamma 2) grid such that the edges of G are straight-line segments. 1 Introduction We consider the problem of embedding the vertices of a planar graph into a small grid in the plane in such a way that the edges are straight, non-intersecting line segments. The existence of such straight-line embeddings for planar graphs was independently discovered by F'ary [Fa48], Stein [St51], and Wagner [Wa36]; this result also follows from Steinitz's theorem on convex polytopes in three dimensions [SR34]. The first algorithms for constructing straightline embeddings [Tu63, CYN84, CON85] required high-precision arithmetic, and the resulting drawings were not very aesthetic, since they tend to produce uneven distributions of vertices over the drawing area. Rosenstiehl and Tarjan [RT86] noticed that it would be convenient to be able to map veritices of a planar graph into a...

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There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c 1 n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embedded in the 1-skeleton of the interior of a compact triangulated orientable PL 3-manifold M . There is a positive constant c2 such that for each t 1, if M consists of t tetrahedra, and K is unknotted, then there is a sequence of at most 2 c 2 t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M . We obtain explicit values for c1 and c2 . Keywords: knot theory, knot diagram, Reidemeister move, normal surfaces, computational complexity This paper grew out of work begun while the authors were visiting the Mathematical Sciences Research Institute in Berkeley in 1996/7. Research at MSRI is supported in part by NSF grant DMS-9022140. The first au...

We give a detailed description of the embedding phase of the Hopcroft and Tarjan planarity testing algorithm. The embedding phase runs in linear time. An implementation based on this paper can be found in [MMN93].