Abstract: We consider the problem of drawing plane graphs with an arbitrarily high vertex degree orthogonally into the plane such that the number of bends on the edges should be minimized. It has been known how to achieve the bend minimum without any restriction of the size of the vertices. Naturally, the vertices should be represented by uniformly small squares. In addition we might require that each face should be represented by a non-empty region. This would allow a labeling of the faces. We present an efficient algorithm which provably achieves the bend minimum following these constraints. Omitting the latter requirement we conjecture that the problem becomes NP-hard. For that case we give advices for good approximations. We demonstrate the effectiveness of our approaches giving some interesting examples.

Abstract: In 1965 Ringel raised a 6 color problem for graphs that can be stated in at least three different forms. In particular, is it possible to color the vertices and faces of every plane graph with 6 colors so that any two adjacent or incident elements are colored differently? This 6 color problem was solved in 1984 by the present author; the proof used about 35 reducible configurations. A shorter new proof is given here using only half as many of reducible configurations. © 1995 John Wiley & Sons, Inc.

Abstract: We consider the special case of the traveling salesman problem (TSP) in which the distance metric is the shortest-path metric of a planar unweighted graph. We present a polynomial-time approximation scheme (PTAS) for this problem.

Abstract: This paper deals with the problem of broadcasting in minimum time in the telephone and message-passing models. Approximation algorithms are developed for arbitrary graphs as well as for several restricted graph classes. In particular, an $O(\sqrt{n})$-additive approximation algorithm is given for broadcasting in general graphs, and an $O(\log n/\log\log n)$ (multiplicative) ratio approximation is given for broadcasting in the open-path model. This also results in an algorithm for broadcasting on random graphs (in the telephone and message-passing model) that yields an $O(\log n/\log\log n)$ approximation with high probability. In addition, the paper presents a broadcast algorithm for graph families with small separators (such as chordal, $k$-outerplanar, bounded-face planar, and series-parallel graphs), with approximation ratio proportional to the separator size times $\log n$. Finally, an efficient approximation algorithm is presented for the class of graphs representable as trees of cliques.

Abstract: We consider the problem of connecting distinguished terminal pairs in a graph via edgedisjoint paths. This is a classical NP-complete problem for which no general approximation techniques are known; it has recently been brought into focus in papers discussing applications to admission control in high-speed networks and to routing in all-optical networks. In this paper we provide O(logn)-approximation algorithms for two natural optimization versions of this problem for the class of nearly-Eulerian, uniformly high-diameter planar graphs, which includes two-dimensional meshes and other common planar interconnection networks. We give an O(logn)-approximation to the maximum number of terminal pairs that can be simultaneously connected via edge-disjoint paths, and an O(logn)-approximation to the minimum number of wavelengths needed to route a collection of terminal pairs in the “optical routing” model considered by Raghavan and Upfal, and others. The latter result improves on an O(log n)-approximation for the special case of the mesh obtained independently by Aumann and Rabani. For both problems the O(logn)-approximation is a consequence of an O(1)-approximation for the special case when all terminal pairs are roughly the same distance apart. Our algorithms make use of a number of new techniques, including the construction of a “crossbar” structure in any nearly-Eulerian planar graph, and develops some connections with classical matroid algorithms.

Abstract: It is shown that a planar graph withouti-circuits, 4 ≤i ≤ 9, is 3-colorable. This result strengthens the result obtained by H.L. Abbott and B. Zhou.

Abstract: Drawing graphs is an important problem that combines elements 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: 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. We present efficient dynamic drawing algorithms for trees and series-parallel digraphs. As further applications of the model, we give dynamic drawing algorithms for planar st-digraphs and planar graphs. Our algorithms adopt a variety of representations (e.g., straight line, polyline, visibility) and update the drawing in a smooth way.

Abstract: The orientable genus is determined for any graph that embeds into the projective plane, Σ, to be essentially half of the representativity of any embedding into Σ. In addition, a structure is given for any 3-connected projective planar graph as the union of a spanning planar graph and a variation of a Mobius Ladder. © 1995 John Wiley & Sons, Inc.

Abstract: A large number of data-parallel applications can be represented as computational graphs from the perspective of parallel computing. The nodes of these graphs represent tasks that can be executed concurrently, while the edges represent the interactions between them. Further, the computational graphs derived from many applications are such that the vertices correspond to multi-dimensional coordinates, and the interaction between computations is limited to vertices that are physically proximate. In this paper we show that graphs with these properties can be transformed into simple architecture-independent representations that encapsulate the locality in these graphs. This representation allows a fast mapping of the computational graph onto the underlying architecture at the time of execution. This is necessary for environments where available computational resources can fre determined only at the time of execution or that change during execution.

Abstract: An orthogonal drawing is an embedding of a graph such that edges are drawn as sequences of horizontal and vertical segments. In this paper we explore lower bounds. We find lower bounds on the number of bends when crossings are allowed, and lower bounds on both the grid-size and the number of bends for planar and plane drawings.

Abstract: In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph such that each path connects two specified vertices on the boundary of the graph. We will focus on the “classical” case where an instance additionally fulfills the so-calledevenness-condition. The fastest algorithm for this problem known from the literature requiresO (n 5/3(loglogn)1/3) time, wheren denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which results in anO(n) algorithm. The proof of correctness immediately yields an alternative proof of the Theorem of Okamura and Seymour, which states a necessary and sufficient condition for solvability.

Abstract: There is much current interest among researchers to find algorithms that will draw graphs in three dimensions. It is well known that every 3-connected planar graph can be represented as a strictly convex polyhedron. However, no practical algorithms exist to draw a general 3-connected planar graph as a convex polyhedron. In this paper we review the concept of a stressed graph and how it relates to convex polyhedra; we present a practical algorithm that uses stressed graphs to draw 3-connected planar graphs as strictly convex polyhedra; and show some examples.

Abstract: 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 (2n3 - 1) x (2n3 - 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)[email protected]?, even if the other one is allowed to be unbounded. In this paper we show that this bound is tight by presenting a grid drawing algorithm that produces drawings of width @?2(n - 1)[email protected]?. The height of the produced drawings is bounded by [email protected]?2(n - 1)[email protected]? - 1. Our algorithm runs in linear time and is easy to implement.

Abstract: The recognition problem for visibility graphs of simple polygons is not known to be in NP, nor is it known to be NP-hard. It is, however, known to be inPSPACE. Further, every such visibility graph can be dismantled as a sequence of visibility graphs of convex fans. Any nondegenerated configuration ofn points can be associated with amaximal chain in the weak Bruhat order of the symmetric groupSn. The visibility graph ofany simple polygon defined on this configuration is completely determined by this maximal chain via a one-to-one correspondence between maximal chains andbalanced tableaux of a certain shape. In the case of staircase polygons (special convex fans), we define a class of graphs calledpersistent graphs and show that the visibility graph of a staircase polygon is persistent. We then describe a polynomial-time algorithm that recovers a representative maximal chain in the weak Bruhat order from a given persistent graph, thus characterizing the class of persistent graphs. The question of recovering a staircase polygon from a given persistent graph, via a maximal chain, is studied in the companion paper [4]. The overall goal of both papers is to offer a characterization of visibility graphs, of convex fans.

Abstract: We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. Our data structures can be updated after any such change in only polylogarithmic time, while a single-pair query is answered in sublinear time. We also describe the first parallel algorithms for solving the dynamic version of the shortest path problem.

Abstract: A planarizing set of a graph is a set of edges or vertices whose removal leaves a planar graph. It is shown that, if G is an n-vertex graph of maximum degree d and orientable genus g, then there exists a planarizing set of O(√dgn) edges. This result is tight within a constant factor. Similar results are obtained for planarizing vertex sets and for graphs embedded on nonorientable surfaces. Planarizing edge and vertex sets can be found in O(n+g) time, if an embedding of G on a surface of genus g is given. We also construct an approximation algorithm that finds an O(√gn log g) planarizing vertex set of G in O(n log g) time if no genus-g embedding is given as an input.

Abstract: Several applications require human interaction during the design process. The user is given the ability to alter the graph as the design progresses. Interactive Graph Drawing gives the user the ability to dynamically interact with the drawing. In this paper we discuss features that are essential for an interactive drawing system. We also describe some possible interactive drawing scenaria and present results on two of them. In these results we assume that the underline drawing is always orthogonal and the maximum degree of any vertex is at most four at the end of any update operation.

Abstract: In this paper we develop a new linear-time algorithm for the recognition of series parallel graphs The algorithm is based on a succinct representation of series parallel graphs for which the presence of an arc can be tested in constant time; space utilization is linear in the number of vertices We show how to compute such a representation in linear time from a breadth-first spanning tree Furthermore, we present a precise condition for the existence of such succinct representations in general, which is, for instance, satisfied by planar graphs

Abstract: In this paper, we study planar directed ordered connected acyclic graphs, in particular graphs that can be built over a (finite) doubly ranked alphabet by parallel and serial composition. On the one hand we introduce finite automata on graphs and, on the other hand, rational expressions that involve union, nondeterministic parallel composition, serial composition, and the iterations of these compositions. We prove a Kleene Theorem linking these two characterizations of sets of graphs.