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The crossing number of a projective graph is quadratic in the facewidth
 ELECTRON J. COMBIN
, 2008
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Approximating the Crossing Number of Graphs Embeddable In Any Orientable Surface
"... The crossing number of a graph is the least number of pairwise edge crossings in a drawing of the graph in the plane. We provide an O(n log n) time constant factor approximation algorithm for the crossing number of a graph of bounded maximum degree which is “densely enough” embeddable in an arbitrar ..."
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Cited by 5 (3 self)
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The crossing number of a graph is the least number of pairwise edge crossings in a drawing of the graph in the plane. We provide an O(n log n) time constant factor approximation algorithm for the crossing number of a graph of bounded maximum degree which is “densely enough” embeddable in an arbitrary fixed orientable surface. Our approach combines some known tools with a powerful new lower bound on the crossing number of an embedded graph. This result extends previous results that gave such approximations in particular cases of projective, toroidal or apex graphs; it is a qualitative improvement over previously published algorithms that constructed lowcrossingnumber drawings of embeddable graphs without giving any approximation guarantees. No constant factor approximation algorithms for the crossing number problem over comparably rich classes of graphs are known to date.
Approximating the Crossing Number of Toroidal Graphs
"... CrossingNumber is one of the most challenging algorithmic problems in topological graph theory, with applications to graph drawing and VLSI layout. No polynomial time constant approximation algorithm is known for this NPcomplete problem. We prove that a natural approach to planar drawing of toroidal ..."
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CrossingNumber is one of the most challenging algorithmic problems in topological graph theory, with applications to graph drawing and VLSI layout. No polynomial time constant approximation algorithm is known for this NPcomplete problem. We prove that a natural approach to planar drawing of toroidal graphs (used already by Pach and Tóth in [20]) gives a polynomial time constant approximation algorithm for the crossing number of toroidal graphs with bounded degree. In this proof we present a new “grid” theorem on toroidal graphs.
On Crossing Numbers of Geometric Proximity Graphs
"... Let P be a set of n points in the plane. A geometric proximity graph on P is a graph where two points are connected by a straightline segment if they satisfy some prescribed proximity rule. We consider four classes of higher order proximity graphs, namely, the knearest neighbor graph, the krelati ..."
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Cited by 3 (2 self)
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Let P be a set of n points in the plane. A geometric proximity graph on P is a graph where two points are connected by a straightline segment if they satisfy some prescribed proximity rule. We consider four classes of higher order proximity graphs, namely, the knearest neighbor graph, the krelative neighborhood graph, the kGabriel graph and the kDelaunay graph. For k = 0 (k = 1 in the case of the knearest neighbor graph) these graphs are plane, but for higher values of k in general they contain crossings. In this paper we provide lower and upper bounds on their minimum and maximum number of crossings. We give general bounds and we also study particular cases that are especially interesting from the viewpoint of applications. These cases include the 1Delaunay graph and the knearest neighbor graph for small values of k.
Optimal Labeling for Connectivity Checking in Planar Networks with Obstacles
, 2009
"... We consider the problem of determining in a planar graph G whether two vertices x and y are linked by a path that avoids a set X of vertices and a set F of edges. We attach labels to vertices in such a way that this fact can be determined from the labels of x and y, the vertices in X and the ends of ..."
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Cited by 3 (3 self)
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We consider the problem of determining in a planar graph G whether two vertices x and y are linked by a path that avoids a set X of vertices and a set F of edges. We attach labels to vertices in such a way that this fact can be determined from the labels of x and y, the vertices in X and the ends of the edges of F. For a planar graph with n vertices, we construct labels of size O(log n). The problem is motivated by the need to quickly compute alternative routes in networks under node or edge failures.
Stars and Bonds in CrossingCritical Graphs
, 2009
"... The structure of previous known infinite families of crossing–critical graphs had led to the conjecture that crossing–critical graphs have bounded bandwidth. If true, this would imply that crossing–critical graphs have bounded degree, that is, that they cannot contain subdivisions of K1,n for arbitr ..."
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The structure of previous known infinite families of crossing–critical graphs had led to the conjecture that crossing–critical graphs have bounded bandwidth. If true, this would imply that crossing–critical graphs have bounded degree, that is, that they cannot contain subdivisions of K1,n for arbitrarily large n. In this paper we prove two new results that revolve around this question. On the positive side, we show that crossing–critical graphs cannot contain subdivisions of K2,n for arbitrarily large n. On the negative side, we show that there are simple 3connected graphs with arbitrarily large maximum degree that are 2crossing–critical in the projective plane. Although the former conjecture is now disproved in a subsequent manuscript by Dvoˇrák and Mohar, our results are not affected, and some interesting questions remain. Namely, can the bandwidth conjecture still be true for simple 3connected graphs in the plane?
Graph Structurings  Some Algorithmic Applications
, 2008
"... Every property definable in monadic second order logic can be checked in polynomialtime on graph classes of bounded cliquewidth. Cliquewidth is a graph parameter defined in an algebraical way, i.e., with operations concatenating graphs and that generalize concatenation of words. Rankwidth, def ..."
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Every property definable in monadic second order logic can be checked in polynomialtime on graph classes of bounded cliquewidth. Cliquewidth is a graph parameter defined in an algebraical way, i.e., with operations concatenating graphs and that generalize concatenation of words. Rankwidth, defined in a combinatorial way, is equivalent to the cliquewidth of undirected graphs. We give an algebraic characterization of rankwidth and we show that rankwidth is linearly bounded in term of treewidth. We also propose a notion of rankwidth of directed graphs and a vertexminor inclusion for directed graphs. We show that directed graphs of bounded rankwidth are characterized by a finite list of finite directed graphs to exclude as vertexminor. Many graph classes do not have bounded rankwidth, e.g., planar graphs. We are interested in labeling schemes on these graph classes. A labeling scheme for a property P in a graph G consists in assigning a label, as short as possible, to each vertex of G and such that, we can verify if G satisfies P by just looking at the labels. We show that every property definable in first order logic admit labeling schemes with labels of logarithmic size on graph classes that have bounded local cliquewidth. Bounded degree graph classes, minor closed classes of graphs
An Algorithm for the Graph Crossing Number Problem
, 2010
"... We study the Minimum Crossing Number problem: given an nvertex graph G, the goal is to find a drawing of G in the plane with minimum number of edge crossings. This is one of the central problems in topological graph theory, that has been studied extensively over the past three decades. The first no ..."
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We study the Minimum Crossing Number problem: given an nvertex graph G, the goal is to find a drawing of G in the plane with minimum number of edge crossings. This is one of the central problems in topological graph theory, that has been studied extensively over the past three decades. The first nontrivial efficient algorithm for the problem, due to Leighton and Rao, achieved an O ( n log 4 n)approximation for bounded degree graphs. This algorithm has since been ( improved by polylogarithmic factors, with the best current approximation ratio standing on O n · poly(d) · log 3/2) n for graphs with maximum degree d. In contrast, only APXhardness is known on the negative side. In this paper we present an efficient randomized algorithm to find a drawing of any nvertex graph G in the plane with O ( OPT 10 · poly(d · log n) ) crossings, where OPT is the number of crossings in the optimal solution, and d is the maximum vertex degree in G. This result implies an Õ ( n9/10 · poly(d) )approximation for Minimum Crossing Number, thus breaking the longstanding Õ(n)approximation barrier for boundeddegree graphs.
On Crossings in Geometric Proximity Graphs
"... We study the number of crossings among edges of some higher order proximity graphs of the family of the Delaunay graph. That is, given a set P of n points in the Euclidean plane, we give lower and upper bounds on the minimum and the maximum number of crossings that these geometric graphs defined on ..."
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We study the number of crossings among edges of some higher order proximity graphs of the family of the Delaunay graph. That is, given a set P of n points in the Euclidean plane, we give lower and upper bounds on the minimum and the maximum number of crossings that these geometric graphs defined on P have. 1