Results 1 
5 of
5
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 ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.
Advances in the Planarization Method: Effective Multiple Edge Insertions
"... Abstract. The planarization method is the strongest known method to heuristically find good solutions to the general crossing number problem in graphs: starting from a planar subgraph, one iteratively inserts edges, representing crossings via dummy nodes. In the recent years, several improvements bo ..."
Abstract
 Add to MetaCart
Abstract. The planarization method is the strongest known method to heuristically find good solutions to the general crossing number problem in graphs: starting from a planar subgraph, one iteratively inserts edges, representing crossings via dummy nodes. In the recent years, several improvements both from the practical and the theoretical point of view have been made. We review these advances and conduct an extensive study of the algorithms ’ practical implications. Thereby, we present the first implementation of an approximation algorithm for the crossing number problem of general graphs, and compare the obtained results with known exact crossing number solutions. 1