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27
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 30 (18 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
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 ..."
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Cited by 25 (5 self)
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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.
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 ..."
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Cited by 20 (8 self)
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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.
Chordal deletion is fixedparameter tractable
 In 32nd International Workshop on GraphTheoretic Concepts in Computer Science, WG 2006, LNCS Proceedings
, 2004
"... Abstract. It is known to be NPhard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomialtime 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 value ..."
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Cited by 19 (1 self)
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Abstract. It is known to be NPhard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomialtime 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 fixedparameter tractable. This answers an open question of Cai [2]. 1
Crossing Number is Hard for Cubic Graphs
"... 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, t ..."
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Cited by 17 (0 self)
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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.
An Efficient Fixed Parameter Tractable Algorithm for 1Sided Crossing Minimization
"... We give an O(OE k \Delta n²) fixed parameter tractable algorithm for the 1Sided Crossing Minimization problem. The constant OE in the running time is the golden ratio OE = ..."
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Cited by 11 (4 self)
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We give an O(OE k \Delta n²) fixed parameter tractable algorithm for the 1Sided Crossing Minimization problem. The constant OE in the running time is the golden ratio OE =
Obtaining a planar graph by vertex deletion
 In Proceedings of the 33rd International Workshop on GraphTheoretic Concepts in Computer Science (WG’07). LNCS Series
"... Abstract. In the kApex problem the task is to find at most k vertices whose deletion makes the given graph planar. The graphs for which there exists a solution form a minor closed class of graphs, hence by the deep results of Robertson and Seymour [31, 30], there is an O(n 3) time algorithm for eve ..."
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Cited by 10 (2 self)
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Abstract. In the kApex problem the task is to find at most k vertices whose deletion makes the given graph planar. The graphs for which there exists a solution form a minor closed class of graphs, hence by the deep results of Robertson and Seymour [31, 30], there is an O(n 3) time algorithm for every fixed value of k. However, the proof is extremely complicated and the constants hidden by the bigO notation are huge. Here we give a much simpler algorithm for this problem with quadratic running time, by iteratively reducing the input graph and then applying techniques for graphs of bounded treewidth.
Parameterized Complexity of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 8 (1 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter intractability results are surveyed as well. Finally, we give some directions for future research.
On brambles, gridlike minors, and parameterized intractability of monadic secondorder logic
"... 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 ..."
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Cited by 6 (1 self)
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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 logfactors. We use this algorithm to construct gridlike minors, a replacement structure for gridminors recently introduced by Reed and Wood, in polynomial time. Using the gridlike