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Planar crossing numbers of graphs embeddable in another
 Internat. J. Found. Comput. Sci
, 2006
"... Let G be a graph of n vertices with maximum degree d that can be drawn without crossing in a closed surface of Euler characteristic χ. It is proved that then G can be drawn in the plane with at most cχdn crossings, where cχ is a constant depending only on χ. This result, which is tight up to a const ..."
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Let G be a graph of n vertices with maximum degree d that can be drawn without crossing in a closed surface of Euler characteristic χ. It is proved that then G can be drawn in the plane with at most cχdn crossings, where cχ is a constant depending only on χ. This result, which is tight up to a constant factor, is strengthened and generalized to the case when there is no restriction on the degrees of the vertices. 1 Introduction and
Inserting a vertex into a planar graph
 In ACMSIAM Symposium on Discrete Algorithms 2009; ACM Press
, 2009
"... We consider the problem of computing a crossing minimum drawing of a given planar graph G = (V, E) augmented by a star, i.e., an additional vertex v together with its incident edges Ev = {(v, u)  u ∈ V}, in which all crossings involve Ev. Alternatively, the problem can be stated as finding a plana ..."
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We consider the problem of computing a crossing minimum drawing of a given planar graph G = (V, E) augmented by a star, i.e., an additional vertex v together with its incident edges Ev = {(v, u)  u ∈ V}, in which all crossings involve Ev. Alternatively, the problem can be stated as finding a planar embedding of G, in which the given star can be inserted requiring the minimum number of crossings. This is a generalization of the crossing minimum edge insertion problem [15], and can help to find improved approximations for the crossing minimization problem. Indeed, in practice, the algorithm for the crossing minimum edge insertion problem turned out to be the key for obtaining the currently strongest approximate solutions for the crossing number of general graphs. The generalization considered here can lead to even better solutions for the crossing minimization problem. Furthermore, it offers new insight into the crossing number problem for almostplanar and apex graphs. It has been an open problem whether the star insertion problem is polynomially solvable. We give an affirmative answer by describing the first efficient algorithm for this problem. This algorithm uses the SPQRtree data structure to handle the exponential number of possible embeddings, in conjunction with dynamic programming schemes for which we introduce partitioning cost subproblems. 1
Vertex insertion approximates the crossing number of apex graphs
 European Journal of Combinatorics
"... Abstract. We prove that the crossing number of an apex graph, i.e. a graph G from which only one vertex v has to be removed to make it planar, can be approximated up to a factor of ∆(G−v)·d(v)/2 by solving the vertex inserting problem, i.e. inserting a vertex plus incident edges into an optimally ch ..."
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Abstract. We prove that the crossing number of an apex graph, i.e. a graph G from which only one vertex v has to be removed to make it planar, can be approximated up to a factor of ∆(G−v)·d(v)/2 by solving the vertex inserting problem, i.e. inserting a vertex plus incident edges into an optimally chosen planar embedding of a planar graph. Since the latter problem can be solved in polynomial time, this establishes the first polynomial fixedfactor approximation algorithm for the crossing number problem of apex graphs with bounded degree. Furthermore, we extend this result by showing that the optimal solution for inserting multiple edges or vertices into a planar graph also approximates the crossing number of the resulting graph.
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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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.
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.
A tighter insertionbased approximation of the crossing number
, 2011
"... Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. As an exact solution to MEI is NPhard for general F, we present the ..."
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Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. As an exact solution to MEI is NPhard for general F, we present the first approximation algorithm for MEI which achieves an additive approximation factor (depending only on the size of F and the maximum degree of G) in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the Falmostplanar graph G+F, while computing the crossing number of G+F exactly is NPhard already when F = 1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of Falmostplanar graphs, achieving constantfactor approximation for the large class of such graphs of bounded degrees and bounded size of F.