Approximating the Crossing Number of Apex Graphs
Chimani, Markus and Hlinený, Petr and Mutzel, Petra (2009) Approximating the Crossing Number of Apex Graphs. In: Graph Drawing 16th International Symposium, GD 2008, September 21- 24, 2008, Heraklion, Crete, Greece , pp. 432-434 (Official URL: http://dx.doi.org/10.1007/978-3-642-00219-9_42).
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We show 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. Due to a recently developed polynomial algorithm for the latter problem, this establishes the ﬁrst polynomial ﬁxed-constant approximation algorithm for the crossing number problem of apex graphs with bounded degree.
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