Abstract not found

In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

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 long-standing 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 second-order arithmetic.

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer 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 crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. 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 so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.

Abstract. It is known to be NP-hard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomial-time algorithm for the problem: the running time is f(k) ·n α for some constant α not depending on k and some f depending only on k. For large values of n, such an algorithm is much better than trying all the O(n k) possibilities. Therefore, the chordal deletion problem parameterized by the number k of vertices to be deleted is fixed-parameter tractable. This answers an open question of Cai [2]. 1

It was proved by [Garey and Johnson, 1983] that computing the crossing number of a graph is an NP-hard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here that it is NP-hard to determine the crossing number of a simple cubic graph. In particular, this implies that the minor-monotone version of crossing number is also NP-hard, which has been open till now.

We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1-SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.

This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixed-parameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixed-parameter intractability results are surveyed as well. Finally, we give some directions for future research.

This thesis is presented in two parts. In Part One we concentrate on algorithmic aspects of parameterized complexity. We explore ways in which the concepts and algorithmic techniques of parameterized complexity can be fruitfully brought to bear on a (classically) well-studied problem area, that of scheduling problems modelled on partial orderings. We develop e#cient and constructive algorithms for parameterized versions of some classically intractable scheduling problems.

Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in |G|. On the positive side, they show the existence of polynomialsized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to log-factors. We use this algorithm to construct grid-like minors, a replacement structure for grid-minors recently introduced by Reed and Wood, in polynomial time. Using the gridlike