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 Hudák (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-vertex graph that admits a 1-planar drawing with straight-line edges has at most 4n-9 edges (Didimo, 2013); and (ii) every drawing of a maximally dense right angle crossing graph is 1-planar (Eades and Liotta, 2013).