Testing gap k-planarity is NP-complete

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