A note on 1-planar graphs
A graph is 1-planar  if it can be drawn in the plane such that each of its edges is crossed at most once. We prove a conjecture of Czap and Hudk (2013) stating that the edge set of every 1-planar graph can be decomposed into a planar graph and a forest. We also provide simple proofs for the following recent results: (i)an n n -vertex graph that admits a 1-planar drawing with straight-line edges has at most 4n?9 4 n ? 9 edges(Didimo, 2013); and (ii)every drawing of a maximally dense right angle crossing graph is 1-planar (Eades and Liotta, 2013).
1-planar graphs right angle crossings