Crossing Number and Weighted Crossing Number of Near-Planar Graphs

A nonplanar graph G is near-planar if it contains an edge e such that G−e is planar. The problem of determining the crossing number of a near-planar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop minmax formulas involving efficiently computable lower and upper bounds. These minmax results are the first of their kind in the study of crossing numbers and improve the approximation factor for the approximation algorithm given by Hlinˇený and Salazar (Graph Drawing GD’06). On the other hand, we show that it is NP-hard to compute a weighted version of the crossing number for near-planar graphs. Keywords Crossing number · Near-planar · Almost planar · Planar separation · Dual distance · Facial distance · NP-hardness

• Type: Journal
• Year: 2011
• Where: ALGORITHMICA
• Authors: Sergio Cabello, Bojan Mohar
• Keywords: ALGORITHMICA 2011, Artificial Intelligence, Minmax Formulas, Near-planar Graphs, Nonplanar Graph