Testing gap k-planarity is NP-complete

Abstract

For all k≥1, we show that deciding whether a graph is k-planar is NP-complete, extending the well-known fact that deciding 1-planarity is NP-complete. Furthermore, we show that the gap version of this decision problem is NP-complete. In particular, given a graph with local crossing number either at most k≥1 or at least 2k, we show that it is NP-complete to decide whether the local crossing number is at most k or at least 2k. This algorithmic lower bound proves the non-existence of a (2−ϵ)-approximation algorithm for any fixed k≥1. In addition, we analyze the sometimes competing relationship between the local crossing number (maximum number of crossings per edge) and crossing number (total number of crossings) of a drawing. We present results regarding the non-existence of drawings that simultaneously approximately minimize both the local crossing number and crossing number of a graph.