Results 11  20
of
187
Recognizing string graphs in NP
 J. of Computer and System Sciences
"... A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP. The recognition problem was not known to be decidable u ..."
Abstract

Cited by 25 (4 self)
 Add to MetaCart
A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP. The recognition problem was not known to be decidable until very recently, when two independent papers established exponential upper bounds on the number of intersections needed to realize a string graph (Pach and Tóth, 2001; Schaefer and ˇ Stefankovič, 2001). These results implied that the recognition problem lies in NEXP. In the present paper we improve this by showing that the recognition problem for string graphs is in NP, and therefore NPcomplete, since Kratochvíl showed that the recognition problem is NPhard (Kratochvíl, 1991b). The result has consequences for the computational complexity of problems in graph drawing, and topological inference. We also show that the string graph problem is decidable for surfaces of arbitrary genus. Key words: String graphs, NPcompleteness, graph drawing, topological inference, Euler diagrams
Toward the Rectilinear Crossing Number of K_n : New Embeddings, Upper Bounds, and Asymptotics
, 2000
"... Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by ..."
Abstract

Cited by 22 (1 self)
 Add to MetaCart
Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear. The rectilinear crossing number of Kn is the fewest number of edge crossings attainable over all planar rectilinear embeddings of Kn . For each n we construct a rectilinear embedding of Kn that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of embeddings of Kn with good asymptotics. Finally, we mention some old and new open problems.
Planar Polyline Drawings with Good Angular Resolution
 Graph Drawing (Proc. GD '98), volume 1547 of LNCS
, 1998
"... . We present a linear time algorithm that constructs a planar polyline grid drawing of any plane graph with n vertices and maximum degree d on a (2n \Gamma 5) \Theta ( 3 2 n \Gamma 7 2 ) grid with at most 5n \Gamma 15 bends and minimum angle ? 2 d . In the constructed drawings, every edge h ..."
Abstract

Cited by 22 (1 self)
 Add to MetaCart
. We present a linear time algorithm that constructs a planar polyline grid drawing of any plane graph with n vertices and maximum degree d on a (2n \Gamma 5) \Theta ( 3 2 n \Gamma 7 2 ) grid with at most 5n \Gamma 15 bends and minimum angle ? 2 d . In the constructed drawings, every edge has at most three bends and length O(n). To our best knowledge, this algorithm achieves the best simultaneous bounds concerning the grid size, angular resolution, and number of bends for planar grid drawings of highdegree planar graphs. Besides the nice theoretical features, the practical drawings are aesthetically very pleasing. An implementation of our algorithm is available with the AGDLibrary (Algorithms for Graph Drawing) [2, 1]. Our algorithm is based on ideas by Kant for polyline grid drawings for triconnected plane graphs [23]. In particular, our algorithm significantly improves upon his bounds on the angular resolution and the grid size for nontriconnected plane graphs....
On the Parameterized Complexity of Layered Graph Drawing
 PROC. 5TH ANNUAL EUROPEAN SYMP. ON ALGORITHMS (ESA '01
, 2001
"... We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for ..."
Abstract

Cited by 21 (9 self)
 Add to MetaCart
We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a lineartime algorithm to decide if a graph has a crossingfree hlayer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossingfree drawing (for fixed k or r). If the number of crossings or deleted edges is a nonfixed parameter then these problems are NPcomplete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the socalled Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.
A Practical Approach to Drawing Undirected Graphs
, 1994
"... Although there is extensive research on drawing graphs, none of the published methods are satisfactory for drawing general undirected graphs. Generating drawings which are optimal with respect to several aesthetic criteria is known to be NPhard, so all published approaches to the problem have used ..."
Abstract

Cited by 21 (2 self)
 Add to MetaCart
Although there is extensive research on drawing graphs, none of the published methods are satisfactory for drawing general undirected graphs. Generating drawings which are optimal with respect to several aesthetic criteria is known to be NPhard, so all published approaches to the problem have used heuristics. These heuristics are too slow to be practical for graphs of moderate size, and they do not produce consistently good drawings for general graphs. Moreover, they rely on general optimization methods, because problemspecific methods require a deeper theoretical understanding of the graph drawing problem. This paper presents an algorithm to generate twodimensional drawings of undirected graphs. The algorithm uses a combination of heuristics to obtain drawings which are nearoptimal with respect to an aesthetic cost function. The algorithm is incremental in nature, but preprocesses the graph to determine an order for node placement. The algorithm uses a local optimization strategy...
Drawing Nice Projections of Objects in Space
, 1995
"... Given a polygonal object (simple polygon, geometric graph, wireframe, skeleton or more generally a set of line segments) in three dimensional Euclidean space, we consider the problem of computing a variety of "nice" parallel (orthographic) projections of the object. We show that given a general pol ..."
Abstract

Cited by 20 (8 self)
 Add to MetaCart
Given a polygonal object (simple polygon, geometric graph, wireframe, skeleton or more generally a set of line segments) in three dimensional Euclidean space, we consider the problem of computing a variety of "nice" parallel (orthographic) projections of the object. We show that given a general polygonal object consisting of n line segments in space, deciding whether it admits a crossingfree projection can be done in O(n 2 log n+k) time and O(n 2 +k) space, where k is the number of edge intersections of forbidden quadrilaterals (i.e. set of directions that admits a crossing) and varies from zero to O(n 4 ). This implies for example that given a simple polygon in 3space we can determine if there exists a plane on which the projection is a simple polygon, within the same complexity. Furthermore, if such a projection does not exist, a minimumcrossing projection can be found in O(n 4 ) time and space. We show that an object always admits a regular projection (of interest to k...
A Polyhedral Approach to the MultiLayer Crossing Minimization Problem
 PROCEEDINGS OF THE INTERNATIONAL SYMPOSIUM ON GRAPH DRAWING, LECTURE NOTES IN COMPUTER SCIENCE 1353
, 1997
"... We study the multilayer crossing minimization problem from a polyhedral point of view. After the introduction of an integer programming formulation of the multilayer crossing minimization problem, we examine the 2layer case and derive several classes of facets of the associated polytope. Prelimin ..."
Abstract

Cited by 20 (2 self)
 Add to MetaCart
We study the multilayer crossing minimization problem from a polyhedral point of view. After the introduction of an integer programming formulation of the multilayer crossing minimization problem, we examine the 2layer case and derive several classes of facets of the associated polytope. Preliminary computational results for 2 and 3layer instances indicate, that the usage of the corresponding facetdefining inequalities in a branchandcut approach may only lead to a practically useful algorithm, if deeper polyhedral studies are conducted.
Using Genetic Algorithms for Drawing Undirected Graphs
 The Third Nordic Workshop on Genetic Algorithms and their Applications
, 1996
"... In this paper we report on our experiences with applying genetic algorithms to the drawing of undirected graphs with straight line edges. Since there exists a relatively simple but powerful heuristic for this class of graphs, namely the spring algorithm, we use this algorithm as a local finetuner w ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
In this paper we report on our experiences with applying genetic algorithms to the drawing of undirected graphs with straight line edges. Since there exists a relatively simple but powerful heuristic for this class of graphs, namely the spring algorithm, we use this algorithm as a local finetuner within the genetic algorithm. We compare our results with drawings produced by the spring algorithm alone and discuss the strengths and weaknesses of the approach presented. Keywords: spring algorithm, evolutionary algorithm, genetic algorithm, graph drawing 1 Introduction The problem of drawing a graph nicely can be regarded as searching for an optimal layout of a given graph according to some measurable aesthetics. However, solving this problem to optimality seems to be computationally infeasible even for relatively simple aesthetic criteria [7], thus one is bound to the area of heuristics and stochastic search methods. In recent years, a lot of research on the problem to support the auto...
Emerging SmallWorld Referral Networks in Evolutionary Labor Markets
 IEEE Transactions on Evolutionary Computation
, 2001
"... We model a labor market that includes referral networks using an agent based simulation. Agents maximize their employment satisfaction by allocating resources to build friendship networks and to adjust searchintensity. We use a local selection evolutionary algorithm, whichmaintains a diverse popula ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
We model a labor market that includes referral networks using an agent based simulation. Agents maximize their employment satisfaction by allocating resources to build friendship networks and to adjust searchintensity. We use a local selection evolutionary algorithm, whichmaintains a diverse population of strategies, to study the adaptive graph topologies resulting from the model. The evolved networks display mixtures of regularity and randomness, as in smallworld networks. A second characteristic emerges in our model as time progresses; the population loses e# ciency due to overcompetition for job referral contacts in away similar to social dilemmas such as the tragedy of the commons. Analysis reveals that the loss of global #tness is driven by an increase in individual robustness, whichallows agents to live longer by surviving job losses. The behavior of our model suggests predictions for a number of policies. Keywords Labor markets, referral networks, local selection, small...
A Numerical Optimization Approach to General Graph Drawing
, 1999
"... 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 ..."
Abstract

Cited by 19 (0 self)
 Add to MetaCart
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