Results 1  10
of
46
Metrics for Graph Drawing Aesthetics
 AUSTRALIAN COMPUTER SOCIETY
, 2001
"... Graph layout algorithms typically conform to one or more aesthetic criteria (e.g. minimising the number of bends, maximising orthogonality). Determining the extent to which a graph drawing conforms to an aesthetic criterion tends to be done informally, and varies between different algorithms. This p ..."
Abstract

Cited by 101 (2 self)
 Add to MetaCart
Graph layout algorithms typically conform to one or more aesthetic criteria (e.g. minimising the number of bends, maximising orthogonality). Determining the extent to which a graph drawing conforms to an aesthetic criterion tends to be done informally, and varies between different algorithms. This paper presents formal metrics for measuring the aesthetic presence in a graph drawing for seven common aesthetic criteria, applicable to any graph drawing of any size. The metrics are useful for determining the aesthetic quality of a given graph drawing, or for defining a cost function for genetic algorithms or simulated annealing programs. The metrics are continuous, so that aesthetic quality is not stated as a binary conformance decision (i.e. the drawing either conforms to the aesthetic or not), but can be stated as the extent of aesthetic conformance using a number between 0 and 1. The paper presents the seven metric formulae. The application of these metrics is demonstrated through the aesthetic analysis of example graph drawings produced by common layout algorithms.
Decidability of String Graphs
 Proceedings of the 33rd Annual Symposium on the Theory of Computing
, 2003
"... We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvl and Matousek [KM91] and settles th ..."
Abstract

Cited by 33 (5 self)
 Add to MetaCart
We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvl and Matousek [KM91] and settles the longstanding open problem of the decidability of string graph recognition (Sinden [Sin66], Graham [Gra76]). Finally we show how to apply the result to solve another old open problem: deciding the existence of Euler diagrams, a fundamental problem of topological inference (Grigni, Papadias, Papadimitriou [GPP95]). The general theory of Euler diagrams turns out to be as hard as secondorder arithmetic.
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 31 (5 self)
 Add to MetaCart
(Show Context)
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
Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs
, 2006
"... Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete a ..."
Abstract

Cited by 27 (1 self)
 Add to MetaCart
Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c> 1024/31827> 0.032. The proof has two new ingredients, interesting in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5(v − 2); and (2) the crossing number of any graph is at least 7 25 e − (v − 2). Both bounds are tight upt o
Crossing Number is Hard for Cubic Graphs
, 2004
"... It was proved by [Garey and Johnson, 1983] that computing the crossing number of a graph is an NPhard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here that it is NPhard to determine the crossing number of a simple cubic graph. In particular, ..."
Abstract

Cited by 22 (0 self)
 Add to MetaCart
It was proved by [Garey and Johnson, 1983] that computing the crossing number of a graph is an NPhard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here that it is NPhard to determine the crossing number of a simple cubic graph. In particular, this implies that the minormonotone version of crossing number is also NPhard, which has been open till now.
CrossingNumber Critical Graphs have Bounded Pathwidth
, 2000
"... . The crossing number of a graph G, denoted by cr(G), is dened as the smallest possible number of edgecrossings in a drawing of G in the plane. A graph G is crossingcritical if cr(G e) < cr(G) for all edges e of G. We prove that crossingcritical graphs have \bounded pathwidth" (by a ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
. The crossing number of a graph G, denoted by cr(G), is dened as the smallest possible number of edgecrossings in a drawing of G in the plane. A graph G is crossingcritical if cr(G e) < cr(G) for all edges e of G. We prove that crossingcritical graphs have \bounded pathwidth" (by a function of the crossing number), which roughly means that such graphs are made up of small pieces joined in a linear way on small cutsets. Equivalently, a crossingcritical graph cannot contain a subdivision of a \large" binary tree. This assertion was conjectured earlier by Salazar in [J. Geelen, B. Richter, G. Salazar, Embedding grids on surfaces, manuscript, 2000]. 1 Introduction We begin with the most important denitions here. Additional denitions and comments will be presented in the subsequent section. If % : [0; 1] ! IR 2 is a simple continuous function, then %([0; 1]) is a simple curve, and %((0; 1)) is a simple open curve. Denition. A graph G is drawn in the plane if the ver...
CHARACTERISATIONS AND EXAMPLES OF GRAPH CLASSES WITH BOUNDED EXPANSION
"... Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of t ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several lineartime algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of socalled topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with ErdösRényi model of random graphs with constant average degree. In particular, we prove that for every fixed d> 0, there exists a class with bounded expansion, such that a random graph of order n and edge probability d/n asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded nonrepetitive chromatic number. We also prove that graphs with ‘linear ’ crossing number are contained in a topologicallyclosed class, while graphs with bounded crossing number are contained in a minorclosed class.